Daily Papers

We prove new convergence rates for a generalized version of stochastic Nesterov acceleration under interpolation conditions. Unlike previous analyses, our approach accelerates any stochastic gradient method which makes sufficient progress in expectation. The proof, which proceeds using the estimating sequences framework, applies to both convex and strongly convex functions and is easily specialized to accelerated SGD under the strong growth condition. In this special case, our analysis reduces the dependence on the strong growth constant from rho to rho as compared to prior work. This improvement is comparable to a square-root of the condition number in the worst case and address criticism that guarantees for stochastic acceleration could be worse than those for SGD.

Conditional diffusion models can create unseen images in various settings, aiding image interpolation. Interpolation in latent spaces is well-studied, but interpolation with specific conditions like text or poses is less understood. Simple approaches, such as linear interpolation in the space of conditions, often result in images that lack consistency, smoothness, and fidelity. To that end, we introduce a novel training-free technique named Attention Interpolation via Diffusion (AID). Our key contributions include 1) proposing an inner/outer interpolated attention layer; 2) fusing the interpolated attention with self-attention to boost fidelity; and 3) applying beta distribution to selection to increase smoothness. We also present a variant, Prompt-guided Attention Interpolation via Diffusion (PAID), that considers interpolation as a condition-dependent generative process. This method enables the creation of new images with greater consistency, smoothness, and efficiency, and offers control over the exact path of interpolation. Our approach demonstrates effectiveness for conceptual and spatial interpolation. Code and demo are available at https://github.com/QY-H00/attention-interpolation-diffusion.

Generative inbetweening aims to generate intermediate frame sequences by utilizing two key frames as input. Although remarkable progress has been made in video generation models, generative inbetweening still faces challenges in maintaining temporal stability due to the ambiguous interpolation path between two key frames. This issue becomes particularly severe when there is a large motion gap between input frames. In this paper, we propose a straightforward yet highly effective Frame-wise Conditions-driven Video Generation (FCVG) method that significantly enhances the temporal stability of interpolated video frames. Specifically, our FCVG provides an explicit condition for each frame, making it much easier to identify the interpolation path between two input frames and thus ensuring temporally stable production of visually plausible video frames. To achieve this, we suggest extracting matched lines from two input frames that can then be easily interpolated frame by frame, serving as frame-wise conditions seamlessly integrated into existing video generation models. In extensive evaluations covering diverse scenarios such as natural landscapes, complex human poses, camera movements and animations, existing methods often exhibit incoherent transitions across frames. In contrast, our FCVG demonstrates the capability to generate temporally stable videos using both linear and non-linear interpolation curves. Our project page and code are available at https://fcvg-inbetween.github.io/.

This paper introduces a LArge-scale Video Interpolation Benchmark (LAVIB) for the low-level video task of Video Frame Interpolation (VFI). LAVIB comprises a large collection of high-resolution videos sourced from the web through an automated pipeline with minimal requirements for human verification. Metrics are computed for each video's motion magnitudes, luminance conditions, frame sharpness, and contrast. The collection of videos and the creation of quantitative challenges based on these metrics are under-explored by current low-level video task datasets. In total, LAVIB includes 283K clips from 17K ultra-HD videos, covering 77.6 hours. Benchmark train, val, and test sets maintain similar video metric distributions. Further splits are also created for out-of-distribution (OOD) challenges, with train and test splits including videos of dissimilar attributes.

Video frame interpolation (VFI) in scenarios with large motion remains challenging due to motion ambiguity between frames. While event cameras can capture high temporal resolution motion information, existing event-based VFI methods struggle with limited training data and complex motion patterns. In this paper, we introduce Event-Guided Video Diffusion Model (EGVD), a novel framework that leverages the powerful priors of pre-trained stable video diffusion models alongside the precise temporal information from event cameras. Our approach features a Multi-modal Motion Condition Generator (MMCG) that effectively integrates RGB frames and event signals to guide the diffusion process, producing physically realistic intermediate frames. We employ a selective fine-tuning strategy that preserves spatial modeling capabilities while efficiently incorporating event-guided temporal information. We incorporate input-output normalization techniques inspired by recent advances in diffusion modeling to enhance training stability across varying noise levels. To improve generalization, we construct a comprehensive dataset combining both real and simulated event data across diverse scenarios. Extensive experiments on both real and simulated datasets demonstrate that EGVD significantly outperforms existing methods in handling large motion and challenging lighting conditions, achieving substantial improvements in perceptual quality metrics (27.4% better LPIPS on Prophesee and 24.1% on BSRGB) while maintaining competitive fidelity measures. Code and datasets available at: https://github.com/OpenImagingLab/EGVD.

One little-explored frontier of image generation and editing is the task of interpolating between two input images, a feature missing from all currently deployed image generation pipelines. We argue that such a feature can expand the creative applications of such models, and propose a method for zero-shot interpolation using latent diffusion models. We apply interpolation in the latent space at a sequence of decreasing noise levels, then perform denoising conditioned on interpolated text embeddings derived from textual inversion and (optionally) subject poses. For greater consistency, or to specify additional criteria, we can generate several candidates and use CLIP to select the highest quality image. We obtain convincing interpolations across diverse subject poses, image styles, and image content, and show that standard quantitative metrics such as FID are insufficient to measure the quality of an interpolation. Code and data are available at https://clintonjwang.github.io/interpolation.

In this paper, we introduce a novel unsupervised network to denoise microscopy videos featured by image sequences captured by a fixed location microscopy camera. Specifically, we propose a DeepTemporal Interpolation method, leveraging a temporal signal filter integrated into the bottom CNN layers, to restore microscopy videos corrupted by unknown noise types. Our unsupervised denoising architecture is distinguished by its ability to adapt to multiple noise conditions without the need for pre-existing noise distribution knowledge, addressing a significant challenge in real-world medical applications. Furthermore, we evaluate our denoising framework using both real microscopy recordings and simulated data, validating our outperforming video denoising performance across a broad spectrum of noise scenarios. Extensive experiments demonstrate that our unsupervised model consistently outperforms state-of-the-art supervised and unsupervised video denoising techniques, proving especially effective for microscopy videos.

Transition videos play a crucial role in media production, enhancing the flow and coherence of visual narratives. Traditional methods like morphing often lack artistic appeal and require specialized skills, limiting their effectiveness. Recent advances in diffusion model-based video generation offer new possibilities for creating transitions but face challenges such as poor inter-frame relationship modeling and abrupt content changes. We propose a novel training-free Transition Video Generation (TVG) approach using video-level diffusion models that addresses these limitations without additional training. Our method leverages Gaussian Process Regression (GPR) to model latent representations, ensuring smooth and dynamic transitions between frames. Additionally, we introduce interpolation-based conditional controls and a Frequency-aware Bidirectional Fusion (FBiF) architecture to enhance temporal control and transition reliability. Evaluations of benchmark datasets and custom image pairs demonstrate the effectiveness of our approach in generating high-quality smooth transition videos. The code are provided in https://sobeymil.github.io/tvg.com.

We analyze surface patches with a corner that is rounded in the sense that the partial derivatives at that point are antiparallel. Sufficient conditions for G^1 smoothness are given, which, up to a certain degenerate case, are also necessary. Further, we investigate curvature integrability and present examples

In this work, we present a variety of novel information-theoretic generalization bounds for learning algorithms, from the supersample setting of Steinke & Zakynthinou (2020)-the setting of the "conditional mutual information" framework. Our development exploits projecting the loss pair (obtained from a training instance and a testing instance) down to a single number and correlating loss values with a Rademacher sequence (and its shifted variants). The presented bounds include square-root bounds, fast-rate bounds, including those based on variance and sharpness, and bounds for interpolating algorithms etc. We show theoretically or empirically that these bounds are tighter than all information-theoretic bounds known to date on the same supersample setting.

We consider in this work quantities that can be obtained as limits of powers of parametrized matrices, for instance the inverse matrix or the logarithm of the determinant. Under the assumption of affine dependence in the parameters, we use the Empirical Interpolation Method (EIM) to derive an approximation for powers of these matrices, from which we derive a nonintrusive approximation for the aforementioned limits. We derive upper bounds of the error made by the obtained formula. Finally, numerical comparisons with classical intrusive and nonintrusive approximation techniques are provided: in the considered test-cases, our algorithm performs well compared to the nonintrusive ones.

We study the robust interpolation problem of arbitrary data distributions supported on a bounded space and propose a two-fold law of robustness. Robust interpolation refers to the problem of interpolating n noisy training data points in R^d by a Lipschitz function. Although this problem has been well understood when the samples are drawn from an isoperimetry distribution, much remains unknown concerning its performance under generic or even the worst-case distributions. We prove a Lipschitzness lower bound Omega(n/p) of the interpolating neural network with p parameters on arbitrary data distributions. With this result, we validate the law of robustness conjecture in prior work by Bubeck, Li, and Nagaraj on two-layer neural networks with polynomial weights. We then extend our result to arbitrary interpolating approximators and prove a Lipschitzness lower bound Omega(n^{1/d}) for robust interpolation. Our results demonstrate a two-fold law of robustness: i) we show the potential benefit of overparametrization for smooth data interpolation when n=poly(d), and ii) we disprove the potential existence of an O(1)-Lipschitz robust interpolating function when n=exp(omega(d)).

This work investigates the use of smooth neural networks for modeling dynamic variations of implicit surfaces under the level set equation (LSE). For this, it extends the representation of neural implicit surfaces to the space-time R^3times R, which opens up mechanisms for continuous geometric transformations. Examples include evolving an initial surface towards general vector fields, smoothing and sharpening using the mean curvature equation, and interpolations of initial conditions. The network training considers two constraints. A data term is responsible for fitting the initial condition to the corresponding time instant, usually R^3 times {0}. Then, a LSE term forces the network to approximate the underlying geometric evolution given by the LSE, without any supervision. The network can also be initialized based on previously trained initial conditions, resulting in faster convergence compared to the standard approach.

Image interpolation based on diffusion models is promising in creating fresh and interesting images. Advanced interpolation methods mainly focus on spherical linear interpolation, where images are encoded into the noise space and then interpolated for denoising to images. However, existing methods face challenges in effectively interpolating natural images (not generated by diffusion models), thereby restricting their practical applicability. Our experimental investigations reveal that these challenges stem from the invalidity of the encoding noise, which may no longer obey the expected noise distribution, e.g., a normal distribution. To address these challenges, we propose a novel approach to correct noise for image interpolation, NoiseDiffusion. Specifically, NoiseDiffusion approaches the invalid noise to the expected distribution by introducing subtle Gaussian noise and introduces a constraint to suppress noise with extreme values. In this context, promoting noise validity contributes to mitigating image artifacts, but the constraint and introduced exogenous noise typically lead to a reduction in signal-to-noise ratio, i.e., loss of original image information. Hence, NoiseDiffusion performs interpolation within the noisy image space and injects raw images into these noisy counterparts to address the challenge of information loss. Consequently, NoiseDiffusion enables us to interpolate natural images without causing artifacts or information loss, thus achieving the best interpolation results.

Video prediction is a challenging task. The quality of video frames from current state-of-the-art (SOTA) generative models tends to be poor and generalization beyond the training data is difficult. Furthermore, existing prediction frameworks are typically not capable of simultaneously handling other video-related tasks such as unconditional generation or interpolation. In this work, we devise a general-purpose framework called Masked Conditional Video Diffusion (MCVD) for all of these video synthesis tasks using a probabilistic conditional score-based denoising diffusion model, conditioned on past and/or future frames. We train the model in a manner where we randomly and independently mask all the past frames or all the future frames. This novel but straightforward setup allows us to train a single model that is capable of executing a broad range of video tasks, specifically: future/past prediction -- when only future/past frames are masked; unconditional generation -- when both past and future frames are masked; and interpolation -- when neither past nor future frames are masked. Our experiments show that this approach can generate high-quality frames for diverse types of videos. Our MCVD models are built from simple non-recurrent 2D-convolutional architectures, conditioning on blocks of frames and generating blocks of frames. We generate videos of arbitrary lengths autoregressively in a block-wise manner. Our approach yields SOTA results across standard video prediction and interpolation benchmarks, with computation times for training models measured in 1-12 days using le 4 GPUs. Project page: https://mask-cond-video-diffusion.github.io ; Code : https://github.com/voletiv/mcvd-pytorch

We consider a biological population whose environment varies periodically in time, exhibiting two very different "seasons" : one is favorable and the other one is unfavorable. For monotone differential models with concave nonlinearities, we address the following question: the system's period being fixed, under what conditions does there exist a critical duration for the unfavorable season? By "critical duration" we mean that above some threshold, the population cannot sustain and extincts, while below this threshold, the system converges to a unique periodic and positive solution. We term this a "sharp seasonal threshold property" (SSTP, for short). Building upon a previous result, we obtain sufficient conditions for SSTP in any dimension and apply our criterion to a two-dimensional model featuring juvenile and adult populations of insects.

We bound the excess risk of interpolating deep linear networks trained using gradient flow. In a setting previously used to establish risk bounds for the minimum ell_2-norm interpolant, we show that randomly initialized deep linear networks can closely approximate or even match known bounds for the minimum ell_2-norm interpolant. Our analysis also reveals that interpolating deep linear models have exactly the same conditional variance as the minimum ell_2-norm solution. Since the noise affects the excess risk only through the conditional variance, this implies that depth does not improve the algorithm's ability to "hide the noise". Our simulations verify that aspects of our bounds reflect typical behavior for simple data distributions. We also find that similar phenomena are seen in simulations with ReLU networks, although the situation there is more nuanced.

In this work, we study the asymptotic behaviour of solutions to the heat equation in exterior domains, i.e., domains which are the complement of a smooth compact set in R^N. Different homogeneous boundary conditions are considered, including Dirichlet, Robin, and Neumann ones. In this second part of our work, we consider the case of bounded initial data and prove that, after some correction term, the solutions become close to the solutions in the whole space and show how complex behaviours appear. We also analyse the case of initial data in L^p with 1<p<infty where all solutions essentially decay to 0 and the convergence rate could be arbitrarily slow.

In this work, we focus on a fractional differential equation in Riesz form discretized by a polynomial B-spline collocation method. For an arbitrary polynomial degree p, we show that the resulting coefficient matrices possess a Toeplitz-like structure. We investigate their spectral properties via their symbol and we prove that, like for second order differential problems, also in this case the given matrices are ill-conditioned both in the low and high frequencies for large p. More precisely, in the fractional scenario the symbol has a single zero at 0 of order α, with α the fractional derivative order that ranges from 1 to 2, and it presents an exponential decay to zero at π for increasing p that becomes faster as α approaches 1. This translates in a mitigated conditioning in the low frequencies and in a deterioration in the high frequencies when compared to second order problems. Furthermore, the derivation of the symbol reveals another similarity of our problem with a classical diffusion problem. Since the entries of the coefficient matrices are defined as evaluations of fractional derivatives of the B-spline basis at the collocation points, we are able to express the central entries of the coefficient matrix as inner products of two fractional derivatives of cardinal B-splines. Finally, we perform a numerical study of the approximation behavior of polynomial B-spline collocation. This study suggests that, in line with non-fractional diffusion problems, the approximation order for smooth solutions in the fractional case is p+2-α for even p, and p+1-α for odd p.

We present a method for computing free-energy differences using thermodynamic integration with a neural network potential that interpolates between two target Hamiltonians. The interpolation is defined at the sample distribution level, and the neural network potential is optimized to match the corresponding equilibrium potential at every intermediate time-step. Once the interpolating potentials and samples are well-aligned, the free-energy difference can be estimated using (neural) thermodynamic integration. To target molecular systems, we simultaneously couple Lennard-Jones and electrostatic interactions and model the rigid-body rotation of molecules. We report accurate results for several benchmark systems: a Lennard-Jones particle in a Lennard-Jones fluid, as well as the insertion of both water and methane solutes in a water solvent at atomistic resolution using a simple three-body neural-network potential.

For the real part of the Cauchy-type integral that is known to be the logarithmic potential of the double layer, a necessary and sufficient condition for the continuous extension to the Ahlfors-regular boundary is established.

Kernel methods are widely used in machine learning, especially for classification problems. However, the theoretical analysis of kernel classification is still limited. This paper investigates the statistical performances of kernel classifiers. With some mild assumptions on the conditional probability eta(x)=P(Y=1mid X=x), we derive an upper bound on the classification excess risk of a kernel classifier using recent advances in the theory of kernel regression. We also obtain a minimax lower bound for Sobolev spaces, which shows the optimality of the proposed classifier. Our theoretical results can be extended to the generalization error of overparameterized neural network classifiers. To make our theoretical results more applicable in realistic settings, we also propose a simple method to estimate the interpolation smoothness of 2eta(x)-1 and apply the method to real datasets.

We study a version of adversarial classification where an adversary is empowered to corrupt data inputs up to some distance varepsilon, using tools from variational analysis. In particular, we describe necessary conditions associated with the optimal classifier subject to such an adversary. Using the necessary conditions, we derive a geometric evolution equation which can be used to track the change in classification boundaries as varepsilon varies. This evolution equation may be described as an uncoupled system of differential equations in one dimension, or as a mean curvature type equation in higher dimension. In one dimension, and under mild assumptions on the data distribution, we rigorously prove that one can use the initial value problem starting from varepsilon=0, which is simply the Bayes classifier, in order to solve for the global minimizer of the adversarial problem for small values of varepsilon. In higher dimensions we provide a similar result, albeit conditional to the existence of regular solutions of the initial value problem. In the process of proving our main results we obtain a result of independent interest connecting the original adversarial problem with an optimal transport problem under no assumptions on whether classes are balanced or not. Numerical examples illustrating these ideas are also presented.

A class of generative models that unifies flow-based and diffusion-based methods is introduced. These models extend the framework proposed in Albergo & Vanden-Eijnden (2023), enabling the use of a broad class of continuous-time stochastic processes called `stochastic interpolants' to bridge any two arbitrary probability density functions exactly in finite time. These interpolants are built by combining data from the two prescribed densities with an additional latent variable that shapes the bridge in a flexible way. The time-dependent probability density function of the stochastic interpolant is shown to satisfy a first-order transport equation as well as a family of forward and backward Fokker-Planck equations with tunable diffusion coefficient. Upon consideration of the time evolution of an individual sample, this viewpoint immediately leads to both deterministic and stochastic generative models based on probability flow equations or stochastic differential equations with an adjustable level of noise. The drift coefficients entering these models are time-dependent velocity fields characterized as the unique minimizers of simple quadratic objective functions, one of which is a new objective for the score of the interpolant density. We show that minimization of these quadratic objectives leads to control of the likelihood for generative models built upon stochastic dynamics, while likelihood control for deterministic dynamics is more stringent. We also discuss connections with other methods such as score-based diffusion models, stochastic localization processes, probabilistic denoising techniques, and rectifying flows. In addition, we demonstrate that stochastic interpolants recover the Schr\"odinger bridge between the two target densities when explicitly optimizing over the interpolant. Finally, algorithmic aspects are discussed and the approach is illustrated on numerical examples.

I examine some analytical properties of a nonlinear reaction-diffusion system that has been used to model the propagation of a wildfire. I establish global-in-time existence and uniqueness of bounded mild solutions to the Cauchy problem for this system given bounded initial data. In particular, this shows that the model does not allow for thermal blow-up. If the initial temperature and fuel density also satisfy certain integrability conditions, the L^2-norms of these global solutions are uniformly bounded in time. Additionally, I use a bootstrap argument to show that small initial temperatures give rise to solutions that decay to zero as time goes to infinity, proving the existence of initial states that do not develop into travelling combustion waves.

With the development of video generation models has advanced significantly in recent years, we adopt large-scale image-to-video diffusion models for video frame interpolation. We present a conditional encoder designed to adapt an image-to-video model for large-motion frame interpolation. To enhance performance, we integrate a dual-branch feature extractor and propose a cross-frame attention mechanism that effectively captures both spatial and temporal information, enabling accurate interpolations of intermediate frames. Our approach demonstrates superior performance on the Fr\'echet Video Distance (FVD) metric when evaluated against other state-of-the-art approaches, particularly in handling large motion scenarios, highlighting advancements in generative-based methodologies.

In an unbounded plane, straight lines are used extensively for mathematical analysis. They are tools of convenience. However, those with high slope values become unbounded at a faster rate than the independent variable. So, straight lines, in this work, are made to be bounded by introducing a parametric nonlinear term that is positive. The straight lines are transformed into bounded nonlinear curves that become unbounded at a much slower rate than the independent variable. This transforming equation can be expressed as a continued fraction of straight lines. The continued fraction is real-valued and converges to the solutions of the transforming equation. Following Euler's method, the continued fraction has been reduced into an infinite series. The usefulness of the bounding nature of continued fraction is demonstrated by solving the problem of image classification. Parameters estimated on the Fashion-MNIST dataset of greyscale images using continued fraction of regression lines have less variance, converge quickly and are more accurate than the linear counterpart. Moreover, this multi-dimensional parametric estimation problem can be expressed on xy- plane using the parameters of the continued fraction and patterns emerge on planar plots.

In this article, we study the consistency of the template estimation with the Fr\'echet mean in quotient spaces. The Fr\'echet mean in quotient spaces is often used when the observations are deformed or transformed by a group action. We show that in most cases this estimator is actually inconsistent. We exhibit a sufficient condition for this inconsistency, which amounts to the folding of the distribution of the noisy template when it is projected to the quotient space. This condition appears to be fulfilled as soon as the support of the noise is large enough. To quantify this inconsistency we provide lower and upper bounds of the bias as a function of the variability (the noise level). This shows that the consistency bias cannot be neglected when the variability increases.

We present VIDIM, a generative model for video interpolation, which creates short videos given a start and end frame. In order to achieve high fidelity and generate motions unseen in the input data, VIDIM uses cascaded diffusion models to first generate the target video at low resolution, and then generate the high-resolution video conditioned on the low-resolution generated video. We compare VIDIM to previous state-of-the-art methods on video interpolation, and demonstrate how such works fail in most settings where the underlying motion is complex, nonlinear, or ambiguous while VIDIM can easily handle such cases. We additionally demonstrate how classifier-free guidance on the start and end frame and conditioning the super-resolution model on the original high-resolution frames without additional parameters unlocks high-fidelity results. VIDIM is fast to sample from as it jointly denoises all the frames to be generated, requires less than a billion parameters per diffusion model to produce compelling results, and still enjoys scalability and improved quality at larger parameter counts.

Recent work in Video Frame Interpolation (VFI) tries to formulate VFI as a diffusion-based conditional image generation problem, synthesizing the intermediate frame given a random noise and neighboring frames. Due to the relatively high resolution of videos, Latent Diffusion Models (LDMs) are employed as the conditional generation model, where the autoencoder compresses images into latent representations for diffusion and then reconstructs images from these latent representations. Such a formulation poses a crucial challenge: VFI expects that the output is deterministically equal to the ground truth intermediate frame, but LDMs randomly generate a diverse set of different images when the model runs multiple times. The reason for the diverse generation is that the cumulative variance (variance accumulated at each step of generation) of generated latent representations in LDMs is large. This makes the sampling trajectory random, resulting in diverse rather than deterministic generations. To address this problem, we propose our unique solution: Frame Interpolation with Consecutive Brownian Bridge Diffusion. Specifically, we propose consecutive Brownian Bridge diffusion that takes a deterministic initial value as input, resulting in a much smaller cumulative variance of generated latent representations. Our experiments suggest that our method can improve together with the improvement of the autoencoder and achieve state-of-the-art performance in VFI, leaving strong potential for further enhancement.

The Reduced Basis Method can be exploited in an efficient way only if the so-called affine dependence assumption on the operator and right-hand side of the considered problem with respect to the parameters is satisfied. When it is not, the Empirical Interpolation Method is usually used to recover this assumption approximately. In both cases, the Reduced Basis Method requires to access and modify the assembly routines of the corresponding computational code, leading to an intrusive procedure. In this work, we derive variants of the EIM algorithm and explain how they can be used to turn the Reduced Basis Method into a nonintrusive procedure. We present examples of aeroacoustic problems solved by integral equations and show how our algorithms can benefit from the linear algebra tools available in the considered code.

The recent surge of interest in comprehensive multimodal models has necessitated the unification of diverse modalities. However, the unification suffers from disparate methodologies. Continuous visual generation necessitates the full-sequence diffusion-based approach, despite its divergence from the autoregressive modeling in the text domain. We posit that autoregressive modeling, i.e., predicting the future based on past deterministic experience, remains crucial in developing both a visual generation model and a potential unified multimodal model. In this paper, we explore an interpolation between the autoregressive modeling and full-parameters diffusion to model visual information. At its core, we present ACDiT, an Autoregressive blockwise Conditional Diffusion Transformer, where the block size of diffusion, i.e., the size of autoregressive units, can be flexibly adjusted to interpolate between token-wise autoregression and full-sequence diffusion. ACDiT is easy to implement, as simple as creating a Skip-Causal Attention Mask (SCAM) during training. During inference, the process iterates between diffusion denoising and autoregressive decoding that can make full use of KV-Cache. We verify the effectiveness of ACDiT on image and video generation tasks. We also demonstrate that benefitted from autoregressive modeling, ACDiT can be seamlessly used in visual understanding tasks despite being trained on the diffusion objective. The analysis of the trade-off between autoregressive modeling and diffusion demonstrates the potential of ACDiT to be used in long-horizon visual generation tasks. These strengths make it promising as the backbone of future unified models.

Conditional text-to-image generation has seen countless recent improvements in terms of quality, diversity and fidelity. Nevertheless, most state-of-the-art models require numerous inference steps to produce faithful generations, resulting in performance bottlenecks for end-user applications. In this paper we introduce Paella, a novel text-to-image model requiring less than 10 steps to sample high-fidelity images, using a speed-optimized architecture allowing to sample a single image in less than 500 ms, while having 573M parameters. The model operates on a compressed & quantized latent space, it is conditioned on CLIP embeddings and uses an improved sampling function over previous works. Aside from text-conditional image generation, our model is able to do latent space interpolation and image manipulations such as inpainting, outpainting, and structural editing. We release all of our code and pretrained models at https://github.com/dome272/Paella

Recent progress in large-scale text-to-video (T2V) and image-to-video (I2V) diffusion models has greatly enhanced video generation, especially in terms of keyframe interpolation. However, current image-to-video diffusion models, while powerful in generating videos from a single conditioning frame, need adaptation for two-frame (start & end) conditioned generation, which is essential for effective bounded interpolation. Unfortunately, existing approaches that fuse temporally forward and backward paths in parallel often suffer from off-manifold issues, leading to artifacts or requiring multiple iterative re-noising steps. In this work, we introduce a novel, bidirectional sampling strategy to address these off-manifold issues without requiring extensive re-noising or fine-tuning. Our method employs sequential sampling along both forward and backward paths, conditioned on the start and end frames, respectively, ensuring more coherent and on-manifold generation of intermediate frames. Additionally, we incorporate advanced guidance techniques, CFG++ and DDS, to further enhance the interpolation process. By integrating these, our method achieves state-of-the-art performance, efficiently generating high-quality, smooth videos between keyframes. On a single 3090 GPU, our method can interpolate 25 frames at 1024 x 576 resolution in just 195 seconds, establishing it as a leading solution for keyframe interpolation.

Geographical, physical, or economic constraints often result in missing traces within seismic data, making the reconstruction of complete seismic data a crucial step in seismic data processing. Traditional methods for seismic data reconstruction require the selection of multiple empirical parameters and struggle to handle large-scale continuous missing data. With the development of deep learning, various neural networks have demonstrated powerful reconstruction capabilities. However, these convolutional neural networks represent a point-to-point reconstruction approach that may not cover the entire distribution of the dataset. Consequently, when dealing with seismic data featuring complex missing patterns, such networks may experience varying degrees of performance degradation. In response to this challenge, we propose a novel diffusion model reconstruction framework tailored for 3D seismic data. To constrain the results generated by the diffusion model, we introduce conditional supervision constraints into the diffusion model, constraining the generated data of the diffusion model based on the input data to be reconstructed. We introduce a 3D neural network architecture into the diffusion model, successfully extending the 2D diffusion model to 3D space. Additionally, we refine the model's generation process by incorporating missing data into the generation process, resulting in reconstructions with higher consistency. Through ablation studies determining optimal parameter values, our method exhibits superior reconstruction accuracy when applied to both field datasets and synthetic datasets, effectively addressing a wide range of complex missing patterns. Our implementation is available at https://github.com/WAL-l/SeisFusion.

Face video restoration (FVR) is a challenging but important problem where one seeks to recover a perceptually realistic face videos from a low-quality input. While diffusion probabilistic models (DPMs) have been shown to achieve remarkable performance for face image restoration, they often fail to preserve temporally coherent, high-quality videos, compromising the fidelity of reconstructed faces. We present a new conditional diffusion framework called FLAIR for FVR. FLAIR ensures temporal consistency across frames in a computationally efficient fashion by converting a traditional image DPM into a video DPM. The proposed conversion uses a recurrent video refinement layer and a temporal self-attention at different scales. FLAIR also uses a conditional iterative refinement process to balance the perceptual and distortion quality during inference. This process consists of two key components: a data-consistency module that analytically ensures that the generated video precisely matches its degraded observation and a coarse-to-fine image enhancement module specifically for facial regions. Our extensive experiments show superiority of FLAIR over the current state-of-the-art (SOTA) for video super-resolution, deblurring, JPEG restoration, and space-time frame interpolation on two high-quality face video datasets.

The field of video generation has made remarkable advancements, yet there remains a pressing need for a clear, systematic recipe that can guide the development of robust and scalable models. In this work, we present a comprehensive study that systematically explores the interplay of model architectures, training recipes, and data curation strategies, culminating in a simple and scalable text-image-conditioned video generation method, named STIV. Our framework integrates image condition into a Diffusion Transformer (DiT) through frame replacement, while incorporating text conditioning via a joint image-text conditional classifier-free guidance. This design enables STIV to perform both text-to-video (T2V) and text-image-to-video (TI2V) tasks simultaneously. Additionally, STIV can be easily extended to various applications, such as video prediction, frame interpolation, multi-view generation, and long video generation, etc. With comprehensive ablation studies on T2I, T2V, and TI2V, STIV demonstrate strong performance, despite its simple design. An 8.7B model with 512 resolution achieves 83.1 on VBench T2V, surpassing both leading open and closed-source models like CogVideoX-5B, Pika, Kling, and Gen-3. The same-sized model also achieves a state-of-the-art result of 90.1 on VBench I2V task at 512 resolution. By providing a transparent and extensible recipe for building cutting-edge video generation models, we aim to empower future research and accelerate progress toward more versatile and reliable video generation solutions.

We introduce the task of arbitrary spatio-temporal video completion, where a video is generated from arbitrary, user-specified patches placed at any spatial location and timestamp, akin to painting on a video canvas. This flexible formulation naturally unifies many existing controllable video generation tasks--including first-frame image-to-video, inpainting, extension, and interpolation--under a single, cohesive paradigm. Realizing this vision, however, faces a fundamental obstacle in modern latent video diffusion models: the temporal ambiguity introduced by causal VAEs, where multiple pixel frames are compressed into a single latent representation, making precise frame-level conditioning structurally difficult. We address this challenge with VideoCanvas, a novel framework that adapts the In-Context Conditioning (ICC) paradigm to this fine-grained control task with zero new parameters. We propose a hybrid conditioning strategy that decouples spatial and temporal control: spatial placement is handled via zero-padding, while temporal alignment is achieved through Temporal RoPE Interpolation, which assigns each condition a continuous fractional position within the latent sequence. This resolves the VAE's temporal ambiguity and enables pixel-frame-aware control on a frozen backbone. To evaluate this new capability, we develop VideoCanvasBench, the first benchmark for arbitrary spatio-temporal video completion, covering both intra-scene fidelity and inter-scene creativity. Experiments demonstrate that VideoCanvas significantly outperforms existing conditioning paradigms, establishing a new state of the art in flexible and unified video generation.

Existing interpolation methods use pre-trained video diffusion priors to generate intermediate frames between sparsely sampled keyframes. In the absence of 3D geometric guidance, these methods struggle to produce plausible results for complex, articulated human motions and offer limited control over the synthesized dynamics. In this paper, we introduce PoseFuse3D Keyframe Interpolator (PoseFuse3D-KI), a novel framework that integrates 3D human guidance signals into the diffusion process for Controllable Human-centric Keyframe Interpolation (CHKI). To provide rich spatial and structural cues for interpolation, our PoseFuse3D, a 3D-informed control model, features a novel SMPL-X encoder that transforms 3D geometry and shape into the 2D latent conditioning space, alongside a fusion network that integrates these 3D cues with 2D pose embeddings. For evaluation, we build CHKI-Video, a new dataset annotated with both 2D poses and 3D SMPL-X parameters. We show that PoseFuse3D-KI consistently outperforms state-of-the-art baselines on CHKI-Video, achieving a 9% improvement in PSNR and a 38% reduction in LPIPS. Comprehensive ablations demonstrate that our PoseFuse3D model improves interpolation fidelity.

We present Envision3D, a novel method for efficiently generating high-quality 3D content from a single image. Recent methods that extract 3D content from multi-view images generated by diffusion models show great potential. However, it is still challenging for diffusion models to generate dense multi-view consistent images, which is crucial for the quality of 3D content extraction. To address this issue, we propose a novel cascade diffusion framework, which decomposes the challenging dense views generation task into two tractable stages, namely anchor views generation and anchor views interpolation. In the first stage, we train the image diffusion model to generate global consistent anchor views conditioning on image-normal pairs. Subsequently, leveraging our video diffusion model fine-tuned on consecutive multi-view images, we conduct interpolation on the previous anchor views to generate extra dense views. This framework yields dense, multi-view consistent images, providing comprehensive 3D information. To further enhance the overall generation quality, we introduce a coarse-to-fine sampling strategy for the reconstruction algorithm to robustly extract textured meshes from the generated dense images. Extensive experiments demonstrate that our method is capable of generating high-quality 3D content in terms of texture and geometry, surpassing previous image-to-3D baseline methods.

Text-to-image diffusion models benefit artists with high-quality image generation. Yet its stochastic nature prevent artists from creating consistent images of the same character. Existing methods try to tackle this challenge and generate consistent content in various ways. However, they either depend on external data or require expensive tuning of the diffusion model. For this issue, we argue that a lightweight but intricate guidance is enough to function. Aiming at this, we lead the way to formalize the objective of consistent generation, derive a clustering-based score function and propose a novel paradigm, OneActor. We design a cluster-conditioned model which incorporates posterior samples to guide the denoising trajectories towards the target cluster. To overcome the overfitting challenge shared by one-shot tuning pipelines, we devise auxiliary components to simultaneously augment the tuning and regulate the inference. This technique is later verified to significantly enhance the content diversity of generated images. Comprehensive experiments show that our method outperforms a variety of baselines with satisfactory character consistency, superior prompt conformity as well as high image quality. And our method is at least 4 times faster than tuning-based baselines. Furthermore, to our best knowledge, we first prove that the semantic space has the same interpolation property as the latent space dose. This property can serve as another promising tool for fine generation control.

Implicit neural representations (INRs) have emerged as a promising approach for video storage and processing, showing remarkable versatility across various video tasks. However, existing methods often fail to fully leverage their representation capabilities, primarily due to inadequate alignment of intermediate features during target frame decoding. This paper introduces a universal boosting framework for current implicit video representation approaches. Specifically, we utilize a conditional decoder with a temporal-aware affine transform module, which uses the frame index as a prior condition to effectively align intermediate features with target frames. Besides, we introduce a sinusoidal NeRV-like block to generate diverse intermediate features and achieve a more balanced parameter distribution, thereby enhancing the model's capacity. With a high-frequency information-preserving reconstruction loss, our approach successfully boosts multiple baseline INRs in the reconstruction quality and convergence speed for video regression, and exhibits superior inpainting and interpolation results. Further, we integrate a consistent entropy minimization technique and develop video codecs based on these boosted INRs. Experiments on the UVG dataset confirm that our enhanced codecs significantly outperform baseline INRs and offer competitive rate-distortion performance compared to traditional and learning-based codecs.

Colloquially speaking, image generation models based upon diffusion processes are frequently said to exhibit "hallucinations," samples that could never occur in the training data. But where do such hallucinations come from? In this paper, we study a particular failure mode in diffusion models, which we term mode interpolation. Specifically, we find that diffusion models smoothly "interpolate" between nearby data modes in the training set, to generate samples that are completely outside the support of the original training distribution; this phenomenon leads diffusion models to generate artifacts that never existed in real data (i.e., hallucinations). We systematically study the reasons for, and the manifestation of this phenomenon. Through experiments on 1D and 2D Gaussians, we show how a discontinuous loss landscape in the diffusion model's decoder leads to a region where any smooth approximation will cause such hallucinations. Through experiments on artificial datasets with various shapes, we show how hallucination leads to the generation of combinations of shapes that never existed. Finally, we show that diffusion models in fact know when they go out of support and hallucinate. This is captured by the high variance in the trajectory of the generated sample towards the final few backward sampling process. Using a simple metric to capture this variance, we can remove over 95% of hallucinations at generation time while retaining 96% of in-support samples. We conclude our exploration by showing the implications of such hallucination (and its removal) on the collapse (and stabilization) of recursive training on synthetic data with experiments on MNIST and 2D Gaussians dataset. We release our code at https://github.com/locuslab/diffusion-model-hallucination.

Recent advances in diffusion-based and controllable video generation have enabled high-quality and temporally coherent video synthesis, laying the groundwork for immersive interactive gaming experiences. However, current methods face limitations in dynamics, generality, long-term consistency, and efficiency, which limit the ability to create various gameplay videos. To address these gaps, we introduce Hunyuan-GameCraft, a novel framework for high-dynamic interactive video generation in game environments. To achieve fine-grained action control, we unify standard keyboard and mouse inputs into a shared camera representation space, facilitating smooth interpolation between various camera and movement operations. Then we propose a hybrid history-conditioned training strategy that extends video sequences autoregressively while preserving game scene information. Additionally, to enhance inference efficiency and playability, we achieve model distillation to reduce computational overhead while maintaining consistency across long temporal sequences, making it suitable for real-time deployment in complex interactive environments. The model is trained on a large-scale dataset comprising over one million gameplay recordings across over 100 AAA games, ensuring broad coverage and diversity, then fine-tuned on a carefully annotated synthetic dataset to enhance precision and control. The curated game scene data significantly improves the visual fidelity, realism and action controllability. Extensive experiments demonstrate that Hunyuan-GameCraft significantly outperforms existing models, advancing the realism and playability of interactive game video generation.

Current audio-driven facial animation methods achieve impressive results for short videos but suffer from error accumulation and identity drift when extended to longer durations. Existing methods attempt to mitigate this through external spatial control, increasing long-term consistency but compromising the naturalness of motion. We propose KeyFace, a novel two-stage diffusion-based framework, to address these issues. In the first stage, keyframes are generated at a low frame rate, conditioned on audio input and an identity frame, to capture essential facial expressions and movements over extended periods of time. In the second stage, an interpolation model fills in the gaps between keyframes, ensuring smooth transitions and temporal coherence. To further enhance realism, we incorporate continuous emotion representations and handle a wide range of non-speech vocalizations (NSVs), such as laughter and sighs. We also introduce two new evaluation metrics for assessing lip synchronization and NSV generation. Experimental results show that KeyFace outperforms state-of-the-art methods in generating natural, coherent facial animations over extended durations, successfully encompassing NSVs and continuous emotions.

Physics-informed neural networks (PINNs) are emerging as popular mesh-free solvers for partial differential equations (PDEs). Recent extensions decompose the domain, applying different PINNs to solve the equation in each subdomain and aligning the solution at the interface of the subdomains. Hence, they can further alleviate the problem complexity, reduce the computational cost, and allow parallelization. However, the performance of the multi-domain PINNs is sensitive to the choice of the interface conditions for solution alignment. While quite a few conditions have been proposed, there is no suggestion about how to select the conditions according to specific problems. To address this gap, we propose META Learning of Interface Conditions (METALIC), a simple, efficient yet powerful approach to dynamically determine the optimal interface conditions for solving a family of parametric PDEs. Specifically, we develop two contextual multi-arm bandit models. The first one applies to the entire training procedure, and online updates a Gaussian process (GP) reward surrogate that given the PDE parameters and interface conditions predicts the solution error. The second one partitions the training into two stages, one is the stochastic phase and the other deterministic phase; we update a GP surrogate for each phase to enable different condition selections at the two stages so as to further bolster the flexibility and performance. We have shown the advantage of METALIC on four bench-mark PDE families.

The Empirical Interpolation Method (EIM) is a greedy procedure that constructs approximate representations of two-variable functions in separated form. In its classical presentation, the two variables play a non-symmetric role. In this work, we give an equivalent definition of the EIM approximation, in which the two variables play symmetric roles. Then, we give a proof for the existence of this approximation, and extend it up to the convergence of the EIM, and for any norm chosen to compute the error in the greedy step. Finally, we introduce a way to compute a separated representation in the case where the number of selected values is different for each variable. In the case of a physical field measured by sensors, this is useful to discard a broken sensor while keeping the information provided by the associated selected field.

We study a discrete model for generalized exchange-driven growth in which the particle exchanged between two clusters is not limited to be of size one. This set of models include as special cases the usual exchange-driven growth system and the coagulation-fragmentation system with binary fragmentation. Under reasonable general condition on the rate coefficients we establish the existence of admissible solutions, meaning solutions that are obtained as appropriate limit of solutions to a finite-dimensional truncation of the infinite-dimensional ODE. For these solutions we prove that, in the class of models we call isolated both the total number of particles and the total mass are conserved, whereas in those models we can non-isolated only the mass is conserved. Additionally, under more restrictive growth conditions for the rate equations we obtain uniqueness of solutions to the initial value problems.

We discuss the numerical solution of initial value problems for varepsilon^2,varphi''+a(x),varphi=0 in the highly oscillatory regime, i.e., with a(x)>0 and 0<varepsilonll 1. We analyze and implement an approximate solution based on the well-known WKB-ansatz. The resulting approximation error is of magnitude O(varepsilon^{N}) where N refers to the truncation order of the underlying asymptotic series. When the optimal truncation order N_{opt} is chosen, the error behaves like O(varepsilon^{-2}exp(-cvarepsilon^{-1})) with some c>0.

We propose a method for generating fly-through videos of a scene, from a single image and a given camera trajectory. We build upon an image-to-video latent diffusion model. We condition its UNet denoiser on the camera trajectory, using four techniques. (1) We condition the UNet's temporal blocks on raw camera extrinsics, similar to MotionCtrl. (2) We use images containing camera rays and directions, similar to CameraCtrl. (3) We reproject the initial image to subsequent frames and use the resulting video as a condition. (4) We use 2D<=>3D transformers to introduce a global 3D representation, which implicitly conditions on the camera poses. We combine all conditions in a ContolNet-style architecture. We then propose a metric that evaluates overall video quality and the ability to preserve details with view changes, which we use to analyze the trade-offs of individual and combined conditions. Finally, we identify an optimal combination of conditions. We calibrate camera positions in our datasets for scale consistency across scenes, and we train our scene exploration model, CamCtrl3D, demonstrating state-of-theart results.

The pursuit of controllability as a higher standard of visual content creation has yielded remarkable progress in customizable image synthesis. However, achieving controllable video synthesis remains challenging due to the large variation of temporal dynamics and the requirement of cross-frame temporal consistency. Based on the paradigm of compositional generation, this work presents VideoComposer that allows users to flexibly compose a video with textual conditions, spatial conditions, and more importantly temporal conditions. Specifically, considering the characteristic of video data, we introduce the motion vector from compressed videos as an explicit control signal to provide guidance regarding temporal dynamics. In addition, we develop a Spatio-Temporal Condition encoder (STC-encoder) that serves as a unified interface to effectively incorporate the spatial and temporal relations of sequential inputs, with which the model could make better use of temporal conditions and hence achieve higher inter-frame consistency. Extensive experimental results suggest that VideoComposer is able to control the spatial and temporal patterns simultaneously within a synthesized video in various forms, such as text description, sketch sequence, reference video, or even simply hand-crafted motions. The code and models will be publicly available at https://videocomposer.github.io.

In molecular dynamics (MD) simulations, rare events, such as protein folding, are typically studied by means of enhanced sampling techniques, most of which rely on the definition of a collective variable (CV) along which the acceleration occurs. Obtaining an expressive CV is crucial, but often hindered by the lack of information about the particular event, e.g., the transition from unfolded to folded conformation. We propose a simulation-free data augmentation strategy using physics-inspired metrics to generate geodesic interpolations resembling protein folding transitions, thereby improving sampling efficiency without true transition state samples. Leveraging interpolation progress parameters, we introduce a regression-based learning scheme for CV models, which outperforms classifier-based methods when transition state data is limited and noisy

We present a straightforward statistical test to detect certain violations of the assumption that the data are Independent and Identically Distributed (IID). The specific form of violation considered is common across real-world applications: whether the examples are ordered in the dataset such that almost adjacent examples tend to have more similar feature values (e.g. due to distributional drift, or attractive interactions between datapoints). Based on a k-Nearest Neighbors estimate, our approach can be used to audit any multivariate numeric data as well as other data types (image, text, audio, etc.) that can be numerically represented, perhaps with model embeddings. Compared with existing methods to detect drift or auto-correlation, our approach is both applicable to more types of data and also able to detect a wider variety of IID violations in practice. Code: https://github.com/cleanlab/cleanlab

Earth system models suffer from various structural and parametric errors in their representation of nonlinear, multi-scale processes, leading to uncertainties in their long-term projections. The effects of many of these errors (particularly those due to fast physics) can be quantified in short-term simulations, e.g., as differences between the predicted and observed states (analysis increments). With the increase in the availability of high-quality observations and simulations, learning nudging from these increments to correct model errors has become an active research area. However, most studies focus on using neural networks, which while powerful, are hard to interpret, are data-hungry, and poorly generalize out-of-distribution. Here, we show the capabilities of Model Error Discovery with Interpretability and Data Assimilation (MEDIDA), a general, data-efficient framework that uses sparsity-promoting equation-discovery techniques to learn model errors from analysis increments. Using two-layer quasi-geostrophic turbulence as the test case, MEDIDA is shown to successfully discover various linear and nonlinear structural/parametric errors when full observations are available. Discovery from spatially sparse observations is found to require highly accurate interpolation schemes. While NNs have shown success as interpolators in recent studies, here, they are found inadequate due to their inability to accurately represent small scales, a phenomenon known as spectral bias. We show that a general remedy, adding a random Fourier feature layer to the NN, resolves this issue enabling MEDIDA to successfully discover model errors from sparse observations. These promising results suggest that with further development, MEDIDA could be scaled up to models of the Earth system and real observations.

We propose Framer for interactive frame interpolation, which targets producing smoothly transitioning frames between two images as per user creativity. Concretely, besides taking the start and end frames as inputs, our approach supports customizing the transition process by tailoring the trajectory of some selected keypoints. Such a design enjoys two clear benefits. First, incorporating human interaction mitigates the issue arising from numerous possibilities of transforming one image to another, and in turn enables finer control of local motions. Second, as the most basic form of interaction, keypoints help establish the correspondence across frames, enhancing the model to handle challenging cases (e.g., objects on the start and end frames are of different shapes and styles). It is noteworthy that our system also offers an "autopilot" mode, where we introduce a module to estimate the keypoints and refine the trajectory automatically, to simplify the usage in practice. Extensive experimental results demonstrate the appealing performance of Framer on various applications, such as image morphing, time-lapse video generation, cartoon interpolation, etc. The code, the model, and the interface will be released to facilitate further research.

Straight line equation y=mx with slope m, when singularly perturbed as ay^3+y=mx with a positive parameter a, results in S-shaped curves or S-curves on a real plane. As arightarrow 0, we get back y=mx which is a cumulative distribution function of a continuous uniform distribution that describes the occurrence of every event in an interval to be equally probable. As arightarrowinfty, the derivative of y has finite support only at y=0 resembling a degenerate distribution. Based on these arguments, in this work, we propose that these S-curves can represent maximum entropy uniform distribution to a zero entropy single value. We also argue that these S-curves are superposable as they are only parametrically nonlinear but fundamentally linear. So far, the superposed forms have been used to capture the patterns of natural systems such as nonlinear dynamics of biological growth and kinetics of enzyme reactions. Here, we attempt to use the S-curve and its superposed form as statistical models. We fit the models on a classical dataset containing flower measurements of iris plants and analyze their usefulness in pattern recognition. Based on these models, we claim that any non-uniform pattern can be represented as a singular perturbation to uniform distribution. However, our parametric estimation procedure have some limitations such as sensitivity to initial conditions depending on the data at hand.

Generative models inspired by dynamical transport of measure -- such as flows and diffusions -- construct a continuous-time map between two probability densities. Conventionally, one of these is the target density, only accessible through samples, while the other is taken as a simple base density that is data-agnostic. In this work, using the framework of stochastic interpolants, we formalize how to couple the base and the target densities. This enables us to incorporate information about class labels or continuous embeddings to construct dynamical transport maps that serve as conditional generative models. We show that these transport maps can be learned by solving a simple square loss regression problem analogous to the standard independent setting. We demonstrate the usefulness of constructing dependent couplings in practice through experiments in super-resolution and in-painting.

This paper studies the problem of recovering cameras from a set of fundamental matrices. A set of fundamental matrices is said to be compatible if a set of cameras exists for which they are the fundamental matrices. We focus on the complete graph, where fundamental matrices for each pair of cameras are given. Previous work has established necessary and sufficient conditions for compatibility as rank and eigenvalue conditions on the n-view fundamental matrix obtained by concatenating the individual fundamental matrices. In this work, we show that the eigenvalue condition is redundant. We provide explicit homogeneous polynomials that describe necessary and sufficient conditions for compatibility in terms of the fundamental matrices and their epipoles. In this direction, we find that quadruple-wise compatibility is enough to ensure global compatibility for any number of cameras. We demonstrate that for four cameras, compatibility is generically described by triple-wise conditions and one additional equation involving all fundamental matrices.

We consider a family of linear systems A_mu alpha=C with system matrix A_mu depending on a parameter mu and for simplicity parameter-independent right-hand side C. These linear systems typically result from the finite-dimensional approximation of a parameter-dependent boundary-value problem. We derive a procedure based on the Empirical Interpolation Method to obtain a separated representation of the system matrix in the form A_muapproxsum_{m}beta_m(mu)A_{mu_m} for some selected values of the parameter. Such a separated representation is in particular useful in the Reduced Basis Method. The procedure is called nonintrusive since it only requires to access the matrices A_{mu_m}. As such, it offers a crucial advantage over existing approaches that instead derive separated representations requiring to enter the code at the level of assembly. Numerical examples illustrate the performance of our new procedure on a simple one-dimensional boundary-value problem and on three-dimensional acoustic scattering problems solved by a boundary element method.

This paper studies the prediction of a target z from a pair of random variables (x,y), where the ground-truth predictor is additive E[z mid x,y] = f_star(x) +g_{star}(y). We study the performance of empirical risk minimization (ERM) over functions f+g, f in F and g in G, fit on a given training distribution, but evaluated on a test distribution which exhibits covariate shift. We show that, when the class F is "simpler" than G (measured, e.g., in terms of its metric entropy), our predictor is more resilient to heterogenous covariate shifts in which the shift in x is much greater than that in y. These results rely on a novel H\"older style inequality for the Dudley integral which may be of independent interest. Moreover, we corroborate our theoretical findings with experiments demonstrating improved resilience to shifts in "simpler" features across numerous domains.

We present an elementary, self-contained proof of Grothendieck's inequality that unifies the real and complex cases and yields both the Krivine and Haagerup bounds, the current best-known explicit bounds for the real and complex Grothendieck constants respectively. This article is intended to be pedagogical, combining and streamlining known ideas of Lindenstrauss--Pe{\l}czy\'nski, Krivine, and Haagerup into a proof that need only univariate calculus, basic complex variables, and a modicum of linear algebra as prerequisites.

This article deals with the following fractional (p,q)-Choquard equation with exponential growth of the form: $varepsilon^{ps}(-Delta)_{p}^{s}u+varepsilon^{qs}(-Delta)_q^su+ Z(x)(|u|^{p-2}u+|u|^{q-2}u)=varepsilon^{mu-N}[|x|^{-mu}*F(u)]f(u) in R^N, where s\in (0,1), \varepsilon>0 is a parameter, 2\leq p=N{s}<q, and 0<\mu<N. The nonlinear function f has an exponential growth at infinity and the continuous potential function Z satisfies suitable natural conditions. With the help of the Ljusternik-Schnirelmann category theory and variational methods, the multiplicity and concentration of positive solutions are obtained for \varepsilon>0$ small enough. In a certain sense, we generalize some previously known results.

In this paper, we investigate the behavior of gradient descent algorithms in physics-informed machine learning methods like PINNs, which minimize residuals connected to partial differential equations (PDEs). Our key result is that the difficulty in training these models is closely related to the conditioning of a specific differential operator. This operator, in turn, is associated to the Hermitian square of the differential operator of the underlying PDE. If this operator is ill-conditioned, it results in slow or infeasible training. Therefore, preconditioning this operator is crucial. We employ both rigorous mathematical analysis and empirical evaluations to investigate various strategies, explaining how they better condition this critical operator, and consequently improve training.

We study the degree of reliability of extrapolation of complex electromagnetic permittivity functions based on their analyticity properties. Given two analytic functions, representing extrapolants of the same experimental data, we examine how much they can differ at an extrapolation point outside of the experimentally accessible frequency band. We give a sharp upper bound on the worst case extrapolation error, in terms of a solution of an integral equation of Fredholm type. We conjecture and give numerical evidence that this bound exhibits a power law precision deterioration as one moves further away from the frequency band containing measurement data.

Generating synthetic datasets for training face recognition models is challenging because dataset generation entails more than creating high fidelity images. It involves generating multiple images of same subjects under different factors (e.g., variations in pose, illumination, expression, aging and occlusion) which follows the real image conditional distribution. Previous works have studied the generation of synthetic datasets using GAN or 3D models. In this work, we approach the problem from the aspect of combining subject appearance (ID) and external factor (style) conditions. These two conditions provide a direct way to control the inter-class and intra-class variations. To this end, we propose a Dual Condition Face Generator (DCFace) based on a diffusion model. Our novel Patch-wise style extractor and Time-step dependent ID loss enables DCFace to consistently produce face images of the same subject under different styles with precise control. Face recognition models trained on synthetic images from the proposed DCFace provide higher verification accuracies compared to previous works by 6.11% on average in 4 out of 5 test datasets, LFW, CFP-FP, CPLFW, AgeDB and CALFW. Code is available at https://github.com/mk-minchul/dcface

In the era of proliferation of large language and image generation models, the phenomenon of "model collapse" refers to the situation whereby as a model is trained recursively on data generated from previous generations of itself over time, its performance degrades until the model eventually becomes completely useless, i.e the model collapses. In this work, we study this phenomenon in the setting of high-dimensional regression and obtain analytic formulae which quantitatively outline this phenomenon in a broad range of regimes. In the special case of polynomial decaying spectral and source conditions, we obtain modified scaling laws which exhibit new crossover phenomena from fast to slow rates. We also propose a simple strategy based on adaptive regularization to mitigate model collapse. Our theoretical results are validated with experiments.

Algorithms for min-max optimization and variational inequalities are often studied under monotonicity assumptions. Motivated by non-monotone machine learning applications, we follow the line of works [Diakonikolas et al., 2021, Lee and Kim, 2021, Pethick et al., 2022, B\"ohm, 2022] aiming at going beyond monotonicity by considering the weaker negative comonotonicity assumption. In particular, we provide tight complexity analyses for the Proximal Point, Extragradient, and Optimistic Gradient methods in this setup, closing some questions on their working guarantees beyond monotonicity.

We investigate optimal control problems governed by the elliptic partial differential equation -Delta u=f subject to Dirichlet boundary conditions on a given domain Omega. The control variable in this setting is the right-hand side f, and the objective is to minimize a cost functional that depends simultaneously on the control f and on the associated state function u. We establish the existence of optimal controls and analyze their qualitative properties by deriving necessary conditions for optimality. In particular, when pointwise constraints of the form alphale flebeta are imposed a priori on the control, we examine situations where a {\it bang-bang} phenomenon arises, that is where the optimal control f assumes only the extremal values alpha and beta. More precisely, the control takes the form f=alpha1_E+beta1_{Omegasetminus E}, thereby placing the problem within the framework of shape optimization. Under suitable assumptions, we further establish certain regularity properties for the optimal sets E. Finally, in the last part of the paper, we present numerical simulations that illustrate our theoretical findings through a selection of representative examples.

In this manuscript we consider the problem of generalized linear estimation on Gaussian mixture data with labels given by a single-index model. Our first result is a sharp asymptotic expression for the test and training errors in the high-dimensional regime. Motivated by the recent stream of results on the Gaussian universality of the test and training errors in generalized linear estimation, we ask ourselves the question: "when is a single Gaussian enough to characterize the error?". Our formula allow us to give sharp answers to this question, both in the positive and negative directions. More precisely, we show that the sufficient conditions for Gaussian universality (or lack of thereof) crucially depend on the alignment between the target weights and the means and covariances of the mixture clusters, which we precisely quantify. In the particular case of least-squares interpolation, we prove a strong universality property of the training error, and show it follows a simple, closed-form expression. Finally, we apply our results to real datasets, clarifying some recent discussion in the literature about Gaussian universality of the errors in this context.

In deep learning, often the training process finds an interpolator (a solution with 0 training loss), but the test loss is still low. This phenomenon, known as benign overfitting, is a major mystery that received a lot of recent attention. One common mechanism for benign overfitting is implicit regularization, where the training process leads to additional properties for the interpolator, often characterized by minimizing certain norms. However, even for a simple sparse linear regression problem y = beta^{*top} x +xi with sparse beta^*, neither minimum ell_1 or ell_2 norm interpolator gives the optimal test loss. In this work, we give a different parametrization of the model which leads to a new implicit regularization effect that combines the benefit of ell_1 and ell_2 interpolators. We show that training our new model via gradient descent leads to an interpolator with near-optimal test loss. Our result is based on careful analysis of the training dynamics and provides another example of implicit regularization effect that goes beyond norm minimization.

Inverse Stefan problem arising in modeling of laser ablation of biomedical tissues is analyzed, where information on the coefficients, heat flux on the fixed boundary, and density of heat sources are missing and must be found along with the temperature and free boundary. Optimal control framework is employed, where the missing data and the free boundary are components of the control vector, and optimality criteria are based on the final moment measurement of the temperature and position of the free boundary. Discretization by finite differences is pursued, and convergence of the discrete optimal control problems to the original problem is proven.

Human visual imagination usually begins with analogies or rough sketches. For example, given an image with a girl playing guitar before a building, one may analogously imagine how it seems like if Iron Man playing guitar before Pyramid in Egypt. Nonetheless, visual condition may not be precisely aligned with the imaginary result indicated by text prompt, and existing layout-controllable text-to-image (T2I) generation models is prone to producing degraded generated results with obvious artifacts. To address this issue, we present a novel T2I generation method dubbed SmartControl, which is designed to modify the rough visual conditions for adapting to text prompt. The key idea of our SmartControl is to relax the visual condition on the areas that are conflicted with text prompts. In specific, a Control Scale Predictor (CSP) is designed to identify the conflict regions and predict the local control scales, while a dataset with text prompts and rough visual conditions is constructed for training CSP. It is worth noting that, even with a limited number (e.g., 1,000~2,000) of training samples, our SmartControl can generalize well to unseen objects. Extensive experiments on four typical visual condition types clearly show the efficacy of our SmartControl against state-of-the-arts. Source code, pre-trained models, and datasets are available at https://github.com/liuxiaoyu1104/SmartControl.

Temporal interpolation often plays a crucial role to learn meaningful representations in dynamic scenes. In this paper, we propose a novel method to train spatiotemporal neural radiance fields of dynamic scenes based on temporal interpolation of feature vectors. Two feature interpolation methods are suggested depending on underlying representations, neural networks or grids. In the neural representation, we extract features from space-time inputs via multiple neural network modules and interpolate them based on time frames. The proposed multi-level feature interpolation network effectively captures features of both short-term and long-term time ranges. In the grid representation, space-time features are learned via four-dimensional hash grids, which remarkably reduces training time. The grid representation shows more than 100 times faster training speed than the previous neural-net-based methods while maintaining the rendering quality. Concatenating static and dynamic features and adding a simple smoothness term further improve the performance of our proposed models. Despite the simplicity of the model architectures, our method achieved state-of-the-art performance both in rendering quality for the neural representation and in training speed for the grid representation.

We consider the problem of approximating a function from L^2 by an element of a given m-dimensional space V_m, associated with some feature map varphi, using evaluations of the function at random points x_1,dots,x_n. After recalling some results on optimal weighted least-squares using independent and identically distributed points, we consider weighted least-squares using projection determinantal point processes (DPP) or volume sampling. These distributions introduce dependence between the points that promotes diversity in the selected features varphi(x_i). We first provide a generalized version of volume-rescaled sampling yielding quasi-optimality results in expectation with a number of samples n = O(mlog(m)), that means that the expected L^2 error is bounded by a constant times the best approximation error in L^2. Also, further assuming that the function is in some normed vector space H continuously embedded in L^2, we further prove that the approximation is almost surely bounded by the best approximation error measured in the H-norm. This includes the cases of functions from L^infty or reproducing kernel Hilbert spaces. Finally, we present an alternative strategy consisting in using independent repetitions of projection DPP (or volume sampling), yielding similar error bounds as with i.i.d. or volume sampling, but in practice with a much lower number of samples. Numerical experiments illustrate the performance of the different strategies.

Video Frame Interpolation aims to recover realistic missing frames between observed frames, generating a high-frame-rate video from a low-frame-rate video. However, without additional guidance, the large motion between frames makes this problem ill-posed. Event-based Video Frame Interpolation (EVFI) addresses this challenge by using sparse, high-temporal-resolution event measurements as motion guidance. This guidance allows EVFI methods to significantly outperform frame-only methods. However, to date, EVFI methods have relied on a limited set of paired event-frame training data, severely limiting their performance and generalization capabilities. In this work, we overcome the limited data challenge by adapting pre-trained video diffusion models trained on internet-scale datasets to EVFI. We experimentally validate our approach on real-world EVFI datasets, including a new one that we introduce. Our method outperforms existing methods and generalizes across cameras far better than existing approaches.

This paper introduces a new parameterization of deep neural networks (both fully-connected and convolutional) with guaranteed ell^2 Lipschitz bounds, i.e. limited sensitivity to input perturbations. The Lipschitz guarantees are equivalent to the tightest-known bounds based on certification via a semidefinite program (SDP). We provide a ``direct'' parameterization, i.e., a smooth mapping from mathbb R^N onto the set of weights satisfying the SDP-based bound. Moreover, our parameterization is complete, i.e. a neural network satisfies the SDP bound if and only if it can be represented via our parameterization. This enables training using standard gradient methods, without any inner approximation or computationally intensive tasks (e.g. projections or barrier terms) for the SDP constraint. The new parameterization can equivalently be thought of as either a new layer type (the sandwich layer), or a novel parameterization of standard feedforward networks with parameter sharing between neighbouring layers. A comprehensive set of experiments on image classification shows that sandwich layers outperform previous approaches on both empirical and certified robust accuracy. Code is available at https://github.com/acfr/LBDN.

Text-to-image diffusion models offer powerful image editing capabilities. To edit real images, many methods rely on the inversion of the image into Gaussian noise. A common approach to invert an image is to gradually add noise to the image, where the noise is determined by reversing the sampling equation. This process has an inherent tradeoff between reconstruction and editability, limiting the editing of challenging images such as highly-detailed ones. Recognizing the reliance of text-to-image models inversion on a text condition, this work explores the importance of the condition choice. We show that a condition that precisely aligns with the input image significantly improves the inversion quality. Based on our findings, we introduce Tight Inversion, an inversion method that utilizes the most possible precise condition -- the input image itself. This tight condition narrows the distribution of the model's output and enhances both reconstruction and editability. We demonstrate the effectiveness of our approach when combined with existing inversion methods through extensive experiments, evaluating the reconstruction accuracy as well as the integration with various editing methods.

We introduce MarDini, a new family of video diffusion models that integrate the advantages of masked auto-regression (MAR) into a unified diffusion model (DM) framework. Here, MAR handles temporal planning, while DM focuses on spatial generation in an asymmetric network design: i) a MAR-based planning model containing most of the parameters generates planning signals for each masked frame using low-resolution input; ii) a lightweight generation model uses these signals to produce high-resolution frames via diffusion de-noising. MarDini's MAR enables video generation conditioned on any number of masked frames at any frame positions: a single model can handle video interpolation (e.g., masking middle frames), image-to-video generation (e.g., masking from the second frame onward), and video expansion (e.g., masking half the frames). The efficient design allocates most of the computational resources to the low-resolution planning model, making computationally expensive but important spatio-temporal attention feasible at scale. MarDini sets a new state-of-the-art for video interpolation; meanwhile, within few inference steps, it efficiently generates videos on par with those of much more expensive advanced image-to-video models.

We introduce a new paradigm for generative modeling built on Continuous Normalizing Flows (CNFs), allowing us to train CNFs at unprecedented scale. Specifically, we present the notion of Flow Matching (FM), a simulation-free approach for training CNFs based on regressing vector fields of fixed conditional probability paths. Flow Matching is compatible with a general family of Gaussian probability paths for transforming between noise and data samples -- which subsumes existing diffusion paths as specific instances. Interestingly, we find that employing FM with diffusion paths results in a more robust and stable alternative for training diffusion models. Furthermore, Flow Matching opens the door to training CNFs with other, non-diffusion probability paths. An instance of particular interest is using Optimal Transport (OT) displacement interpolation to define the conditional probability paths. These paths are more efficient than diffusion paths, provide faster training and sampling, and result in better generalization. Training CNFs using Flow Matching on ImageNet leads to consistently better performance than alternative diffusion-based methods in terms of both likelihood and sample quality, and allows fast and reliable sample generation using off-the-shelf numerical ODE solvers.

Classifier-free guidance (CFG) is a fundamental tool in modern diffusion models for text-guided generation. Although effective, CFG has notable drawbacks. For instance, DDIM with CFG lacks invertibility, complicating image editing; furthermore, high guidance scales, essential for high-quality outputs, frequently result in issues like mode collapse. Contrary to the widespread belief that these are inherent limitations of diffusion models, this paper reveals that the problems actually stem from the off-manifold phenomenon associated with CFG, rather than the diffusion models themselves. More specifically, inspired by the recent advancements of diffusion model-based inverse problem solvers (DIS), we reformulate text-guidance as an inverse problem with a text-conditioned score matching loss, and develop CFG++, a novel approach that tackles the off-manifold challenges inherent in traditional CFG. CFG++ features a surprisingly simple fix to CFG, yet it offers significant improvements, including better sample quality for text-to-image generation, invertibility, smaller guidance scales, reduced mode collapse, etc. Furthermore, CFG++ enables seamless interpolation between unconditional and conditional sampling at lower guidance scales, consistently outperforming traditional CFG at all scales. Experimental results confirm that our method significantly enhances performance in text-to-image generation, DDIM inversion, editing, and solving inverse problems, suggesting a wide-ranging impact and potential applications in various fields that utilize text guidance. Project Page: https://cfgpp-diffusion.github.io/.

Text-driven large scene image synthesis has made significant progress with diffusion models, but controlling it is challenging. While using additional spatial controls with corresponding texts has improved the controllability of large scene synthesis, it is still challenging to faithfully reflect detailed text descriptions without user-provided controls. Here, we propose DetText2Scene, a novel text-driven large-scale image synthesis with high faithfulness, controllability, and naturalness in a global context for the detailed human-centric text description. Our DetText2Scene consists of 1) hierarchical keypoint-box layout generation from the detailed description by leveraging large language model (LLM), 2) view-wise conditioned joint diffusion process to synthesize a large scene from the given detailed text with LLM-generated grounded keypoint-box layout and 3) pixel perturbation-based pyramidal interpolation to progressively refine the large scene for global coherence. Our DetText2Scene significantly outperforms prior arts in text-to-large scene synthesis qualitatively and quantitatively, demonstrating strong faithfulness with detailed descriptions, superior controllability, and excellent naturalness in a global context.

In large scale machine learning, random sampling is a popular way to approximate datasets by a small representative subset of examples. In particular, sensitivity sampling is an intensely studied technique which provides provable guarantees on the quality of approximation, while reducing the number of examples to the product of the VC dimension d and the total sensitivity mathfrak S in remarkably general settings. However, guarantees going beyond this general bound of mathfrak S d are known in perhaps only one setting, for ell_2 subspace embeddings, despite intense study of sensitivity sampling in prior work. In this work, we show the first bounds for sensitivity sampling for ell_p subspace embeddings for pneq 2 that improve over the general mathfrak S d bound, achieving a bound of roughly mathfrak S^{2/p} for 1leq p<2 and mathfrak S^{2-2/p} for 2<p<infty. For 1leq p<2, we show that this bound is tight, in the sense that there exist matrices for which mathfrak S^{2/p} samples is necessary. Furthermore, our techniques yield further new results in the study of sampling algorithms, showing that the root leverage score sampling algorithm achieves a bound of roughly d for 1leq p<2, and that a combination of leverage score and sensitivity sampling achieves an improved bound of roughly d^{2/p}mathfrak S^{2-4/p} for 2<p<infty. Our sensitivity sampling results yield the best known sample complexity for a wide class of structured matrices that have small ell_p sensitivity.

Backward compatibility of model predictions is a desired property when updating a machine learning driven application. It allows to seamlessly improve the underlying model without introducing regression bugs. In classification tasks these bugs occur in the form of negative flips. This means an instance that was correctly classified by the old model is now classified incorrectly by the updated model. This has direct negative impact on the user experience of such systems e.g. a frequently used voice assistant query is suddenly misclassified. A common reason to update the model is when new training data becomes available and needs to be incorporated. Simply retraining the model with the updated data introduces the unwanted negative flips. We study the problem of regression during data updates and propose Backward Compatible Weight Interpolation (BCWI). This method interpolates between the weights of the old and new model and we show in extensive experiments that it reduces negative flips without sacrificing the improved accuracy of the new model. BCWI is straight forward to implement and does not increase inference cost. We also explore the use of importance weighting during interpolation and averaging the weights of multiple new models in order to further reduce negative flips.

We develop a framework for non-asymptotic analysis of deterministic samplers used for diffusion generative modeling. Several recent works have analyzed stochastic samplers using tools like Girsanov's theorem and a chain rule variant of the interpolation argument. Unfortunately, these techniques give vacuous bounds when applied to deterministic samplers. We give a new operational interpretation for deterministic sampling by showing that one step along the probability flow ODE can be expressed as two steps: 1) a restoration step that runs gradient ascent on the conditional log-likelihood at some infinitesimally previous time, and 2) a degradation step that runs the forward process using noise pointing back towards the current iterate. This perspective allows us to extend denoising diffusion implicit models to general, non-linear forward processes. We then develop the first polynomial convergence bounds for these samplers under mild conditions on the data distribution.

Graphic design plays a vital role in visual communication across advertising, marketing, and multimedia entertainment. Prior work has explored automated graphic design generation using diffusion models, aiming to streamline creative workflows and democratize design capabilities. However, complex graphic design scenarios require accurately adhering to design intent specified by multiple heterogeneous user-provided elements (\eg images, layouts, and texts), which pose multi-condition control challenges for existing methods. Specifically, previous single-condition control models demonstrate effectiveness only within their specialized domains but fail to generalize to other conditions, while existing multi-condition methods often lack fine-grained control over each sub-condition and compromise overall compositional harmony. To address these limitations, we introduce CreatiDesign, a systematic solution for automated graphic design covering both model architecture and dataset construction. First, we design a unified multi-condition driven architecture that enables flexible and precise integration of heterogeneous design elements with minimal architectural modifications to the base diffusion model. Furthermore, to ensure that each condition precisely controls its designated image region and to avoid interference between conditions, we propose a multimodal attention mask mechanism. Additionally, we develop a fully automated pipeline for constructing graphic design datasets, and introduce a new dataset with 400K samples featuring multi-condition annotations, along with a comprehensive benchmark. Experimental results show that CreatiDesign outperforms existing models by a clear margin in faithfully adhering to user intent.

We present Compound Conditioned ControlNet, C3Net, a novel generative neural architecture taking conditions from multiple modalities and synthesizing multimodal contents simultaneously (e.g., image, text, audio). C3Net adapts the ControlNet architecture to jointly train and make inferences on a production-ready diffusion model and its trainable copies. Specifically, C3Net first aligns the conditions from multi-modalities to the same semantic latent space using modality-specific encoders based on contrastive training. Then, it generates multimodal outputs based on the aligned latent space, whose semantic information is combined using a ControlNet-like architecture called Control C3-UNet. Correspondingly, with this system design, our model offers an improved solution for joint-modality generation through learning and explaining multimodal conditions instead of simply taking linear interpolations on the latent space. Meanwhile, as we align conditions to a unified latent space, C3Net only requires one trainable Control C3-UNet to work on multimodal semantic information. Furthermore, our model employs unimodal pretraining on the condition alignment stage, outperforming the non-pretrained alignment even on relatively scarce training data and thus demonstrating high-quality compound condition generation. We contribute the first high-quality tri-modal validation set to validate quantitatively that C3Net outperforms or is on par with first and contemporary state-of-the-art multimodal generation. Our codes and tri-modal dataset will be released.

The higher-order correlation clustering problem is an expressive model, and recently, local search heuristics have been proposed for several applications. Certifying optimality, however, is NP-hard and practically hampered already by the complexity of the problem statement. Here, we focus on establishing partial optimality conditions for the special case of complete graphs and cubic objective functions. In addition, we define and implement algorithms for testing these conditions and examine their effect numerically, on two datasets.

Diffusion models excel at joint pixel sampling for image generation but lack efficient training-free methods for partial conditional sampling (e.g., inpainting with known pixels). Prior work typically formulates this as an intractable inverse problem, relying on coarse variational approximations, heuristic losses requiring expensive backpropagation, or slow stochastic sampling. These limitations preclude: (1) accurate distributional matching in inpainting results, (2) efficient inference modes without gradient, (3) compatibility with fast ODE-based samplers. To address these limitations, we propose LanPaint: a training-free, asymptotically exact partial conditional sampling methods for ODE-based and rectified flow diffusion models. By leveraging carefully designed Langevin dynamics, LanPaint enables fast, backpropagation-free Monte Carlo sampling. Experiments demonstrate that our approach achieves superior performance with precise partial conditioning and visually coherent inpainting across diverse tasks.

We consider a Schr\"odinger-Poisson system involving a general nonlinearity at critical growth and we prove the existence of positive solutions. The Ambrosetti-Rabinowitz condition is not required. We also study the asymptotics of solutions with respect to a parameter.

We give an improved theoretical analysis of score-based generative modeling. Under a score estimate with small L^2 error (averaged across timesteps), we provide efficient convergence guarantees for any data distribution with second-order moment, by either employing early stopping or assuming smoothness condition on the score function of the data distribution. Our result does not rely on any log-concavity or functional inequality assumption and has a logarithmic dependence on the smoothness. In particular, we show that under only a finite second moment condition, approximating the following in reverse KL divergence in epsilon-accuracy can be done in tilde Oleft(d log (1/delta){epsilon}right) steps: 1) the variance-delta Gaussian perturbation of any data distribution; 2) data distributions with 1/delta-smooth score functions. Our analysis also provides a quantitative comparison between different discrete approximations and may guide the choice of discretization points in practice.

In this paper we provide a new method to certify that a nearby polynomial system has a singular isolated root with a prescribed multiplicity structure. More precisely, given a polynomial system f =(f_1, ldots, f_N)in C[x_1, ldots, x_n]^N, we present a Newton iteration on an extended deflated system that locally converges, under regularity conditions, to a small deformation of f such that this deformed system has an exact singular root. The iteration simultaneously converges to the coordinates of the singular root and the coefficients of the so called inverse system that describes the multiplicity structure at the root. We use $alpha$-theory test to certify the quadratic convergence, and togive bounds on the size of the deformation and on the approximation error. The approach relies on an analysis of the punctual Hilbert scheme, for which we provide a new description. We show in particular that some of its strata can be rationally parametrized and exploit these parametrizations in the certification. We show in numerical experimentation how the approximate inverse system can be computed as a starting point of the Newton iterations and the fast numerical convergence to the singular root with its multiplicity structure, certified by our criteria.

We study first-order methods with preconditioning for solving structured nonlinear convex optimization problems. We propose a new family of preconditioners generated by symmetric polynomials. They provide first-order optimization methods with a provable improvement of the condition number, cutting the gaps between highest eigenvalues, without explicit knowledge of the actual spectrum. We give a stochastic interpretation of this preconditioning in terms of coordinate volume sampling and compare it with other classical approaches, including the Chebyshev polynomials. We show how to incorporate a polynomial preconditioning into the Gradient and Fast Gradient Methods and establish the corresponding global complexity bounds. Finally, we propose a simple adaptive search procedure that automatically chooses the best possible polynomial preconditioning for the Gradient Method, minimizing the objective along a low-dimensional Krylov subspace. Numerical experiments confirm the efficiency of our preconditioning strategies for solving various machine learning problems.

In this paper, we provide a finite random iterated function system satisfying the open set condition, for which the random version of Bowen's formula fails to hold. This counterexample shows that analogous results established for random recursive constructions are not always obtained for random iterated function systems.

The reduced basis method is a model reduction technique yielding substantial savings of computational time when a solution to a parametrized equation has to be computed for many values of the parameter. Certification of the approximation is possible by means of an a posteriori error bound. Under appropriate assumptions, this error bound is computed with an algorithm of complexity independent of the size of the full problem. In practice, the evaluation of the error bound can become very sensitive to round-off errors. We propose herein an explanation of this fact. A first remedy has been proposed in [F. Casenave, Accurate a posteriori error evaluation in the reduced basis method. C. R. Math. Acad. Sci. Paris 350 (2012) 539--542.]. Herein, we improve this remedy by proposing a new approximation of the error bound using the Empirical Interpolation Method (EIM). This method achieves higher levels of accuracy and requires potentially less precomputations than the usual formula. A version of the EIM stabilized with respect to round-off errors is also derived. The method is illustrated on a simple one-dimensional diffusion problem and a three-dimensional acoustic scattering problem solved by a boundary element method.

The curvature of ODE trajectories in diffusion models hinders their ability to generate high-quality images in a few number of function evaluations (NFE). In this paper, we propose a novel and effective approach to reduce trajectory curvature by utilizing adaptive conditions. By employing a extremely light-weight quantized encoder, our method incurs only an additional 1% of training parameters, eliminates the need for extra regularization terms, yet achieves significantly better sample quality. Our approach accelerates ODE sampling while preserving the downstream task image editing capabilities of SDE techniques. Extensive experiments verify that our method can generate high quality results under extremely limited sampling costs. With only 6 NFE, we achieve 5.14 FID on CIFAR-10, 6.91 FID on FFHQ 64x64 and 3.10 FID on AFHQv2.

This paper addresses a regression problem in which output label values are the results of sensing the magnitude of a phenomenon. A low value of such labels can mean either that the actual magnitude of the phenomenon was low or that the sensor made an incomplete observation. This leads to a bias toward lower values in labels and the resultant learning because labels may have lower values due to incomplete observations, even if the actual magnitude of the phenomenon was high. Moreover, because an incomplete observation does not provide any tags indicating incompleteness, we cannot eliminate or impute them. To address this issue, we propose a learning algorithm that explicitly models incomplete observations corrupted with an asymmetric noise that always has a negative value. We show that our algorithm is unbiased as if it were learned from uncorrupted data that does not involve incomplete observations. We demonstrate the advantages of our algorithm through numerical experiments.

In the field of portrait video generation, the use of single images to generate portrait videos has become increasingly prevalent. A common approach involves leveraging generative models to enhance adapters for controlled generation. However, control signals (e.g., text, audio, reference image, pose, depth map, etc.) can vary in strength. Among these, weaker conditions often struggle to be effective due to interference from stronger conditions, posing a challenge in balancing these conditions. In our work on portrait video generation, we identified audio signals as particularly weak, often overshadowed by stronger signals such as facial pose and reference image. However, direct training with weak signals often leads to difficulties in convergence. To address this, we propose V-Express, a simple method that balances different control signals through the progressive training and the conditional dropout operation. Our method gradually enables effective control by weak conditions, thereby achieving generation capabilities that simultaneously take into account the facial pose, reference image, and audio. The experimental results demonstrate that our method can effectively generate portrait videos controlled by audio. Furthermore, a potential solution is provided for the simultaneous and effective use of conditions of varying strengths.

In astrophysical and cosmological analyses, the increasing quality and volume of astronomical data demand efficient and precise computational tools. This work introduces a novel adaptive algorithm for automatic knots (AutoKnots) allocation in spline interpolation, designed to meet user-defined precision requirements. Unlike traditional methods that rely on manually configured knot distributions with numerous parameters, the proposed technique automatically determines the optimal number and placement of knots based on interpolation error criteria. This simplifies configuration, often requiring only a single parameter. The algorithm progressively improves the interpolation by adaptively sampling the function-to-be-approximated, f(x), in regions where the interpolation error exceeds the desired threshold. All function evaluations contribute directly to the final approximation, ensuring efficiency. While each resampling step involves recomputing the interpolation table, this process is highly optimized and usually computationally negligible compared to the cost of evaluating f(x). We show the algorithm's efficacy through a series of precision tests on different functions. However, the study underscores the necessity for caution when dealing with certain function types, notably those featuring plateaus. To address this challenge, a heuristic enhancement is incorporated, improving accuracy in flat regions. This algorithm has been extensively used and tested over the years. NumCosmo includes a comprehensive set of unit tests that rigorously evaluate the algorithm both directly and indirectly, underscoring its robustness and reliability. As a practical application, we compute the surface mass density Sigma(R) and the average surface mass density Sigma(<R) for Navarro-Frenk-White and Hernquist halo density profiles, which provide analytical benchmarks. (abridged)

Modern techniques for physical simulations rely on numerical schemes and mesh-refinement methods to address trade-offs between precision and complexity, but these handcrafted solutions are tedious and require high computational power. Data-driven methods based on large-scale machine learning promise high adaptivity by integrating long-range dependencies more directly and efficiently. In this work, we focus on fluid dynamics and address the shortcomings of a large part of the literature, which are based on fixed support for computations and predictions in the form of regular or irregular grids. We propose a novel setup to perform predictions in a continuous spatial and temporal domain while being trained on sparse observations. We formulate the task as a double observation problem and propose a solution with two interlinked dynamical systems defined on, respectively, the sparse positions and the continuous domain, which allows to forecast and interpolate a solution from the initial condition. Our practical implementation involves recurrent GNNs and a spatio-temporal attention observer capable of interpolating the solution at arbitrary locations. Our model not only generalizes to new initial conditions (as standard auto-regressive models do) but also performs evaluation at arbitrary space and time locations. We evaluate on three standard datasets in fluid dynamics and compare to strong baselines, which are outperformed both in classical settings and in the extended new task requiring continuous predictions.

The sufficiently scattered condition (SSC) is a key condition in the study of identifiability of various matrix factorization problems, including nonnegative, minimum-volume, symmetric, simplex-structured, and polytopic matrix factorizations. The SSC allows one to guarantee that the computed matrix factorization is unique/identifiable, up to trivial ambiguities. However, this condition is NP-hard to check in general. In this paper, we show that it can however be checked in a reasonable amount of time in realistic scenarios, when the factorization rank is not too large. This is achieved by formulating the problem as a non-convex quadratic optimization problem over a bounded set. We use the global non-convex optimization software Gurobi, and showcase the usefulness of this code on synthetic data sets and on real-world hyperspectral images.

Virtual sensing techniques allow for inferring signals at new unmonitored locations by exploiting spatio-temporal measurements coming from physical sensors at different locations. However, as the sensor coverage becomes sparse due to costs or other constraints, physical proximity cannot be used to support interpolation. In this paper, we overcome this challenge by leveraging dependencies between the target variable and a set of correlated variables (covariates) that can frequently be associated with each location of interest. From this viewpoint, covariates provide partial observability, and the problem consists of inferring values for unobserved channels by exploiting observations at other locations to learn how such variables can correlate. We introduce a novel graph-based methodology to exploit such relationships and design a graph deep learning architecture, named GgNet, implementing the framework. The proposed approach relies on propagating information over a nested graph structure that is used to learn dependencies between variables as well as locations. GgNet is extensively evaluated under different virtual sensing scenarios, demonstrating higher reconstruction accuracy compared to the state-of-the-art.

Gravitational-wave approximants are essential for gravitational-wave astronomy, allowing the coverage binary black hole parameter space for inference or match filtering without costly numerical relativity (NR) simulations, but generally trading some accuracy for computational efficiency. To reduce this trade-off, NR surrogate models can be constructed using interpolation within NR waveform space. We present a 2-stage training approach for neural network-based NR surrogate models. Initially trained on approximant-generated waveforms and then fine-tuned with NR data, these dual-stage artificial neural surrogate (DANSur) models offer rapid and competitively accurate waveform generation, generating millions in under 20ms on a GPU while keeping mean mismatches with NR around 10^{-4}. Implemented in the bilby framework, we show they can be used for parameter estimation tasks.

Automatic differentiation (autodiff) has revolutionized machine learning. It allows to express complex computations by composing elementary ones in creative ways and removes the burden of computing their derivatives by hand. More recently, differentiation of optimization problem solutions has attracted widespread attention with applications such as optimization layers, and in bi-level problems such as hyper-parameter optimization and meta-learning. However, so far, implicit differentiation remained difficult to use for practitioners, as it often required case-by-case tedious mathematical derivations and implementations. In this paper, we propose automatic implicit differentiation, an efficient and modular approach for implicit differentiation of optimization problems. In our approach, the user defines directly in Python a function F capturing the optimality conditions of the problem to be differentiated. Once this is done, we leverage autodiff of F and the implicit function theorem to automatically differentiate the optimization problem. Our approach thus combines the benefits of implicit differentiation and autodiff. It is efficient as it can be added on top of any state-of-the-art solver and modular as the optimality condition specification is decoupled from the implicit differentiation mechanism. We show that seemingly simple principles allow to recover many existing implicit differentiation methods and create new ones easily. We demonstrate the ease of formulating and solving bi-level optimization problems using our framework. We also showcase an application to the sensitivity analysis of molecular dynamics.

Offline optimization has recently emerged as an increasingly popular approach to mitigate the prohibitively expensive cost of online experimentation. The key idea is to learn a surrogate of the black-box function that underlines the target experiment using a static (offline) dataset of its previous input-output queries. Such an approach is, however, fraught with an out-of-distribution issue where the learned surrogate becomes inaccurate outside the offline data regimes. To mitigate this, existing offline optimizers have proposed numerous conditioning techniques to prevent the learned surrogate from being too erratic. Nonetheless, such conditioning strategies are often specific to particular surrogate or search models, which might not generalize to a different model choice. This motivates us to develop a model-agnostic approach instead, which incorporates a notion of model sharpness into the training loss of the surrogate as a regularizer. Our approach is supported by a new theoretical analysis demonstrating that reducing surrogate sharpness on the offline dataset provably reduces its generalized sharpness on unseen data. Our analysis extends existing theories from bounding generalized prediction loss (on unseen data) with loss sharpness to bounding the worst-case generalized surrogate sharpness with its empirical estimate on training data, providing a new perspective on sharpness regularization. Our extensive experimentation on a diverse range of optimization tasks also shows that reducing surrogate sharpness often leads to significant improvement, marking (up to) a noticeable 9.6% performance boost. Our code is publicly available at https://github.com/cuong-dm/IGNITE

This paper proposes a method for the automatic creation of variables (in the case of regression) that complement the information contained in the initial input vector. The method works as a pre-processing step in which the continuous values of the variable to be regressed are discretized into a set of intervals which are then used to define value thresholds. Then classifiers are trained to predict whether the value to be regressed is less than or equal to each of these thresholds. The different outputs of the classifiers are then concatenated in the form of an additional vector of variables that enriches the initial vector of the regression problem. The implemented system can thus be considered as a generic pre-processing tool. We tested the proposed enrichment method with 5 types of regressors and evaluated it in 33 regression datasets. Our experimental results confirm the interest of the approach.

Singular limits for the following indirect signalling chemotaxis system align* \left\{ array{lllllll} \partial_t n = \Delta n - \nabla \cdot (n \nabla c ) & in \Omega\times(0,\infty) , \varepsilon \partial_t c = \Delta c - c + w & in \Omega\times(0,\infty), \varepsilon \partial_t w = \tau \Delta w - w + n & in \Omega\times (0,\infty), \partial_\nu n = \partial_\nu c = \partial_\nu w = 0, &on \partial\Omega\times (0,\infty) %(n,c,w)_{t=0} = (n_0,c_0,w_0) & on \Omega, array \right. align* are investigated. More precisely, we study parabolic-elliptic simplification, or PES, varepsilonto 0^+ with fixed tau>0 up to the critical dimension N=4, and indirect-direct simplification, or IDS, (varepsilon,tau)to (0^+,0^+) up to the critical dimension N=2. These are relevant in biological situations where the signalling process is on a much faster time scale compared to the species diffusion and all interactions. Showing singular limits in critical dimensions is challenging. To deal with the PES, we carefully combine the entropy function, an Adam-type inequality, the regularisation of slow evolution, and an energy equation method to obtain strong convergence in representative spaces. For the IDS, a bootstrap argument concerning the L^p-energy function is devised, which allows us to obtain suitable uniform bounds for the singular limits. Moreover, in both scenarios, we also present the convergence rates, where the effect of the initial layer and the convergence to the critical manifold are also revealed.

The paper is a continuation of the study started in Yorzh1. Schrodinger operators on finite compact metric graphs are considered under the assumption that the matching conditions at the graph vertices are of delta type. Either an infinite series of trace formulae (provided that edge potentials are infinitely smooth) or a finite number of such formulae (in the cases of L_1 and C^M edge potentials) are obtained which link together two different quantum graphs under the assumption that their spectra coincide. Applications are given to the problem of recovering matching conditions for a quantum graph based on its spectrum.

We extend Textual Inversion to learn pseudo-words that represent a concept at different resolutions. This allows us to generate images that use the concept with different levels of detail and also to manipulate different resolutions using language. Once learned, the user can generate images at different levels of agreement to the original concept; "A photo of S^*(0)" produces the exact object while the prompt "A photo of S^*(0.8)" only matches the rough outlines and colors. Our framework allows us to generate images that use different resolutions of an image (e.g. details, textures, styles) as separate pseudo-words that can be composed in various ways. We open-soure our code in the following URL: https://github.com/giannisdaras/multires_textual_inversion

Lebesgue curve is a space-filling curve that fills the unit square through linear interpolation. In this study, we generalise Lebesgue's construction to generate space-filling curves from any given planar substitution satisfying a mild condition. The generated space-filling curves for some known substitutions are elucidated. Some of those substitutions further induce relatively dense fractal-like sets in the plane, whenever some additional assumptions are met.

We present an online post-hoc calibration method, called Online Platt Scaling (OPS), which combines the Platt scaling technique with online logistic regression. We demonstrate that OPS smoothly adapts between i.i.d. and non-i.i.d. settings with distribution drift. Further, in scenarios where the best Platt scaling model is itself miscalibrated, we enhance OPS by incorporating a recently developed technique called calibeating to make it more robust. Theoretically, our resulting OPS+calibeating method is guaranteed to be calibrated for adversarial outcome sequences. Empirically, it is effective on a range of synthetic and real-world datasets, with and without distribution drifts, achieving superior performance without hyperparameter tuning. Finally, we extend all OPS ideas to the beta scaling method.

This work puts forth a new optimal design formulation for planar elastic membranes. The goal is to minimize the membrane's compliance through choosing the material distribution described by a positive Radon measure. The deformation of the membrane itself is governed by the convexified F\"{o}ppl's model. The uniqueness of this model lies in the convexity of its variational formulation despite the inherent nonlinearity of the strain-displacement relation. It makes it possible to rewrite the optimization problem as a pair of mutually dual convex variational problems. In the primal problem a linear functional is maximized with respect to displacement functions while enforcing that point-wisely the strain lies in an unbounded closed convex set. The dual problem consists in finding equilibrated stresses that are to minimize a convex integral functional of linear growth defined on the space of Radon measures. The pair of problems is analysed: existence and regularity results are provided, together with the system of optimality criteria. To demonstrate the computational potential of the pair, a finite element scheme is developed around it. Upon reformulation to a conic-quadratic & semi-definite programming problem, the method is employed to produce numerical simulations for several load case scenarios.

We present a new distribution-free conformal prediction algorithm for sequential data (e.g., time series), called the sequential predictive conformal inference (SPCI). We specifically account for the nature that time series data are non-exchangeable, and thus many existing conformal prediction algorithms are not applicable. The main idea is to adaptively re-estimate the conditional quantile of non-conformity scores (e.g., prediction residuals), upon exploiting the temporal dependence among them. More precisely, we cast the problem of conformal prediction interval as predicting the quantile of a future residual, given a user-specified point prediction algorithm. Theoretically, we establish asymptotic valid conditional coverage upon extending consistency analyses in quantile regression. Using simulation and real-data experiments, we demonstrate a significant reduction in interval width of SPCI compared to other existing methods under the desired empirical coverage.

We study uncertainty quantification for partial differential equations subject to domain uncertainty. We parameterize the random domain using the model recently considered by Chernov and Le (2024) as well as Harbrecht, Schmidlin, and Schwab (2024) in which the input random field is assumed to belong to a Gevrey smoothness class. This approach has the advantage of being substantially more general than models which assume a particular parametric representation of the input random field such as a Karhunen-Loeve series expansion. We consider both the Poisson equation as well as the heat equation and design randomly shifted lattice quasi-Monte Carlo (QMC) cubature rules for the computation of the expected solution under domain uncertainty. We show that these QMC rules exhibit dimension-independent, essentially linear cubature convergence rates in this framework. In addition, we complete the error analysis by taking into account the approximation errors incurred by dimension truncation of the random input field and finite element discretization. Numerical experiments are presented to confirm the theoretical rates.

Existing calibration methods occasionally fail for large field-of-view cameras due to the non-linearity of the underlying problem and the lack of good initial values for all parameters of the used camera model. This might occur because a simpler projection model is assumed in an initial step, or a poor initial guess for the internal parameters is pre-defined. A lot of the difficulties of general camera calibration lie in the use of a forward projection model. We side-step these challenges by first proposing a solver to calibrate the parameters in terms of a back-projection model and then regress the parameters for a target forward model. These steps are incorporated in a robust estimation framework to cope with outlying detections. Extensive experiments demonstrate that our approach is very reliable and returns the most accurate calibration parameters as measured on the downstream task of absolute pose estimation on test sets. The code is released at https://github.com/ylochman/babelcalib.

We present a generalizable novel view synthesis method which enables modifying the visual appearance of an observed scene so rendered views match a target weather or lighting condition without any scene specific training or access to reference views at the target condition. Our method is based on a pretrained generalizable transformer architecture and is fine-tuned on synthetically generated scenes under different appearance conditions. This allows for rendering novel views in a consistent manner for 3D scenes that were not included in the training set, along with the ability to (i) modify their appearance to match the target condition and (ii) smoothly interpolate between different conditions. Experiments on real and synthetic scenes show that our method is able to generate 3D consistent renderings while making realistic appearance changes, including qualitative and quantitative comparisons. Please refer to our project page for video results: https://ava-nvs.github.io/

The lack of freely available standardized datasets represents an aggravating factor during the development and testing the performance of novel computational techniques in exposure assessment and dosimetry research. This hinders progress as researchers are required to generate numerical data (field, power and temperature distribution) anew using simulation software for each exposure scenario. Other than being time consuming, this approach is highly susceptible to errors that occur during the configuration of the electromagnetic model. To address this issue, in this paper, the limited available data on the incident power density and resultant maximum temperature rise on the skin surface considering various steady-state exposure scenarios at 10-90 GHz have been statistically modeled. The synthetic data have been sampled from the fitted statistical multivariate distribution with respect to predetermined dosimetric constraints. We thus present a comprehensive and open-source dataset compiled of the high-fidelity numerical data considering various exposures to a realistic source. Furthermore, different surrogate models for predicting maximum temperature rise on the skin surface were fitted based on the synthetic dataset. All surrogate models were tested on the originally available data where satisfactory predictive performance has been demonstrated. A simple technique of combining quadratic polynomial and tensor-product spline surrogates, each operating on its own cluster of data, has achieved the lowest mean absolute error of 0.058 {\deg}C. Therefore, overall experimental results indicate the validity of the proposed synthetic dataset.

Generative models have demonstrated remarkable success in domains such as text, image, and video synthesis. In this work, we explore the application of generative models to fluid dynamics, specifically for turbulence simulation, where classical numerical solvers are computationally expensive. We propose a novel stochastic generative model based on stochastic interpolants, which enables probabilistic forecasting while incorporating physical constraints such as energy stability and divergence-freeness. Unlike conventional stochastic generative models, which are often agnostic to underlying physical laws, our approach embeds energy consistency by making the parameters of the stochastic interpolant learnable coefficients. We evaluate our method on a benchmark turbulence problem - Kolmogorov flow - demonstrating superior accuracy and stability over state-of-the-art alternatives such as autoregressive conditional diffusion models (ACDMs) and PDE-Refiner. Furthermore, we achieve stable results for significantly longer roll-outs than standard stochastic interpolants. Our results highlight the potential of physics-aware generative models in accelerating and enhancing turbulence simulations while preserving fundamental conservation properties.

We examine the connections between deterministic, complete, and general global optimisation of continuous functions and a general concept of regression from the perspective of constructive type theory via the concept of 'searchability'. We see how the property of convergence of global optimisation is a straightforward consequence of searchability. The abstract setting allows us to generalise searchability and continuity to higher-order functions, so that we can formulate novel convergence criteria for regression, derived from the convergence of global optimisation. All the theory and the motivating examples are fully formalised in the proof assistant Agda.

Solving linear systems of equations is a common problem that arises both on its own and as a subroutine in more complex problems: given a matrix A and a vector b, find a vector x such that Ax=b. We consider the case where one doesn't need to know the solution x itself, but rather an approximation of the expectation value of some operator associated with x, e.g., x'Mx for some matrix M. In this case, when A is sparse, N by N and has condition number kappa, classical algorithms can find x and estimate x'Mx in O(N sqrt(kappa)) time. Here, we exhibit a quantum algorithm for this task that runs in poly(log N, kappa) time, an exponential improvement over the best classical algorithm.

The interdiffusion coefficients are estimated either following the Wagner's method expressed with respect to the composition (mol or atomic fraction) normalized variable after considering the molar volume variation or the den Broeder's method expressed with respect to the concentration (composition divided by the molar volume) normalized variable. On the other hand, the relations for estimation of the intrinsic diffusion coefficients of components as established by van Loo and integrated diffusion coefficients in a phase with narrow homogeneity range as established by Wagner are currently available with respect to the composition normalized variable only. In this study, we have first derived the relation proposed by den Broeder following the line of treatment proposed by Wagner. Further, the relations for estimation of the intrinsic diffusion coefficients of the components and integrated interdiffusion coefficient are established with respect to the concentration normalized variable, which were not available earlier. The veracity of these methods is examined based on the estimation of data in Ni-Pd, Ni-Al and Cu-Sn systems. Our analysis indicates that both the approaches are logically correct and there is small difference in the estimated data in these systems although a higher difference could be found in other systems. The integrated interdiffusion coefficients with respect to the concentration (or concentration normalized variable) can only be estimated considering the ideal molar volume variation. This might be drawback in certain practical systems.

We show that there is a general, informative and reliable procedure for discovering causal relations when, for all the investigator knows, both latent variables and selection bias may be at work. Given information about conditional independence and dependence relations between measured variables, even when latent variables and selection bias may be present, there are sufficient conditions for reliably concluding that there is a causal path from one variable to another, and sufficient conditions for reliably concluding when no such causal path exists.

Motivated by recent advances in operational weather forecasting, we study the efficacy of low-precision arithmetic for climate simulations. We develop a framework to measure rounding error in a climate model which provides a stress-test for a low-precision version of the model, and we apply our method to a variety of models including the Lorenz system; a shallow water approximation for flow over a ridge; and a coarse resolution global atmospheric model with simplified parameterisations (SPEEDY). Although double precision (52 significant bits) is standard across operational climate models, in our experiments we find that single precision (23 sbits) is more than enough and that as low as half precision (10 sbits) is often sufficient. For example, SPEEDY can be run with 12 sbits across the entire code with negligible rounding error and this can be lowered to 10 sbits if very minor errors are accepted, amounting to less than 0.1 mm/6hr for the average grid-point precipitation, for example. Our test is based on the Wasserstein metric and this provides stringent non-parametric bounds on rounding error accounting for annual means as well as extreme weather events. In addition, by testing models using both round-to-nearest (RN) and stochastic rounding (SR) we find that SR can mitigate rounding error across a range of applications. Thus our results also provide evidence that SR could be relevant to next-generation climate models. While many studies have shown that low-precision arithmetic can be suitable on short-term weather forecasting timescales, our results give the first evidence that a similar low precision level can be suitable for climate.

Many clustering schemes are defined by optimizing an objective function defined on the partitions of the underlying set of a finite metric space. In this paper, we construct a framework for studying what happens when we instead impose various structural conditions on the clustering schemes, under the general heading of functoriality. Functoriality refers to the idea that one should be able to compare the results of clustering algorithms as one varies the data set, for example by adding points or by applying functions to it. We show that within this framework, one can prove a theorems analogous to one of J. Kleinberg, in which for example one obtains an existence and uniqueness theorem instead of a non-existence result. We obtain a full classification of all clustering schemes satisfying a condition we refer to as excisiveness. The classification can be changed by varying the notion of maps of finite metric spaces. The conditions occur naturally when one considers clustering as the statistical version of the geometric notion of connected components. By varying the degree of functoriality that one requires from the schemes it is possible to construct richer families of clustering schemes that exhibit sensitivity to density.

Variational inequalities have recently attracted considerable interest in machine learning as a flexible paradigm for models that go beyond ordinary loss function minimization (such as generative adversarial networks and related deep learning systems). In this setting, the optimal O(1/t) convergence rate for solving smooth monotone variational inequalities is achieved by the Extra-Gradient (EG) algorithm and its variants. Aiming to alleviate the cost of an extra gradient step per iteration (which can become quite substantial in deep learning applications), several algorithms have been proposed as surrogates to Extra-Gradient with a single oracle call per iteration. In this paper, we develop a synthetic view of such algorithms, and we complement the existing literature by showing that they retain a O(1/t) ergodic convergence rate in smooth, deterministic problems. Subsequently, beyond the monotone deterministic case, we also show that the last iterate of single-call, stochastic extra-gradient methods still enjoys a O(1/t) local convergence rate to solutions of non-monotone variational inequalities that satisfy a second-order sufficient condition.

Previous literature offers limited clues on how to learn a periodic function using modern neural networks. We start with a study of the extrapolation properties of neural networks; we prove and demonstrate experimentally that the standard activations functions, such as ReLU, tanh, sigmoid, along with their variants, all fail to learn to extrapolate simple periodic functions. We hypothesize that this is due to their lack of a "periodic" inductive bias. As a fix of this problem, we propose a new activation, namely, x + sin^2(x), which achieves the desired periodic inductive bias to learn a periodic function while maintaining a favorable optimization property of the ReLU-based activations. Experimentally, we apply the proposed method to temperature and financial data prediction.

We define a probabilistic programming language for Gaussian random variables with a first-class exact conditioning construct. We give operational, denotational and equational semantics for this language, establishing convenient properties like exchangeability of conditions. Conditioning on equality of continuous random variables is nontrivial, as the exact observation may have probability zero; this is Borel's paradox. Using categorical formulations of conditional probability, we show that the good properties of our language are not particular to Gaussians, but can be derived from universal properties, thus generalizing to wider settings. We define the Cond construction, which internalizes conditioning as a morphism, providing general compositional semantics for probabilistic programming with exact conditioning.

We study the impact on mechanisms for facility location of moving from one dimension to two (or more) dimensions and Euclidean or Manhattan distances. We consider three fundamental axiomatic properties: anonymity which is a basic fairness property, Pareto optimality which is one of the most important efficiency properties, and strategy proofness which ensures agents do not have an incentive to mis-report. We also consider how well such mechanisms can approximate the optimal welfare. Our results are somewhat negative. Moving from one dimension to two (or more) dimensions often makes these axiomatic properties more difficult to achieve. For example, with two facilities in Euclidean space or with just a single facility in Manhattan space, no mechanism is anonymous, Pareto optimal and strategy proof. By contrast, mechanisms on the line exist with all three properties.We also show that approximation ratios may increase when moving to two (or more) dimensions. All our impossibility results are minimal. If we drop one of the three axioms (anonymity, Pareto optimality or strategy proofness) multiple mechanisms satisfy the other two axioms.

Recent progress in panoramic image generation has underscored two critical limitations in existing approaches. First, most methods are built upon diffusion models, which are inherently ill-suited for equirectangular projection (ERP) panoramas due to the violation of the identically and independently distributed (i.i.d.) Gaussian noise assumption caused by their spherical mapping. Second, these methods often treat text-conditioned generation (text-to-panorama) and image-conditioned generation (panorama outpainting) as separate tasks, relying on distinct architectures and task-specific data. In this work, we propose a unified framework, Panoramic AutoRegressive model (PAR), which leverages masked autoregressive modeling to address these challenges. PAR avoids the i.i.d. assumption constraint and integrates text and image conditioning into a cohesive architecture, enabling seamless generation across tasks. To address the inherent discontinuity in existing generative models, we introduce circular padding to enhance spatial coherence and propose a consistency alignment strategy to improve generation quality. Extensive experiments demonstrate competitive performance in text-to-image generation and panorama outpainting tasks while showcasing promising scalability and generalization capabilities.

We propose an algorithm for inexpensive gradient-based hyperparameter optimization that combines the implicit function theorem (IFT) with efficient inverse Hessian approximations. We present results about the relationship between the IFT and differentiating through optimization, motivating our algorithm. We use the proposed approach to train modern network architectures with millions of weights and millions of hyper-parameters. For example, we learn a data-augmentation network - where every weight is a hyperparameter tuned for validation performance - outputting augmented training examples. Jointly tuning weights and hyperparameters with our approach is only a few times more costly in memory and compute than standard training.

We prove a new existence theorem for proper solutions of Huisken and Ilmanen's weak inverse mean curvature flow, assuming certain non-degeneracy conditions on the isoperimetric profile. In particular, no curvature assumption is imposed in our existence theorem.

We made a comparative analysis of numerical methods for multidimensional optimization. The main parameter is a number of computations of the test function to reach necessary accuracy, as it is computationally "slow". For complex functions, analytic differentiation by many parameters can cause problems associated with a significant complication of the program and thus slowing its operation. For comparison, we used the methods: "brute force" (or minimization on a regular grid), Monte Carlo, steepest descent, conjugate gradients, Brent's method (golden section search), parabolic interpolation etc. The Monte-Carlo method was applied to the eclipsing binary system AM Leo.

Data Augmentation (DA), \ie, synthesizing faithful and diverse samples to expand the original training set, is a prevalent and effective strategy to improve various visual recognition tasks. With the powerful image generation ability, diffusion-based DA has shown strong performance gains on different benchmarks. In this paper, we analyze today's diffusion-based DA methods, and argue that they cannot take account of both faithfulness and diversity, which are two critical keys for generating high-quality samples and boosting final classification performance. To this end, we propose a novel Diffusion-based Inversion Interpolation DA method: Diff-II. Specifically, Diff-II consists of three main steps: 1) Category concepts learning: Learning concept embeddings for each category. 2) Inversion interpolation: Calculating the inversion for each image, and conducting spherical interpolation for two randomly sampled inversions from the same category. 3) Two-stage denoising: Using different prompts to generate synthesized images in a coarse-to-fine manner. Extensive experiments on multiple image classification tasks (\eg, few-shot, long-tailed, and out-of-distribution classification) have demonstrated its effectiveness over state-of-the-art diffusion-based DA methods.

In this paper, by introducing Generalized Bernstein condition, we propose the first Obig(sqrt{p}{nepsilon}big) high probability excess population risk bound for differentially private algorithms under the assumptions G-Lipschitz, L-smooth, and Polyak-{\L}ojasiewicz condition, based on gradient perturbation method. If we replace the properties G-Lipschitz and L-smooth by alpha-H{\"o}lder smoothness (which can be used in non-smooth setting), the high probability bound comes to Obig(n^{-alpha{1+2alpha}}big) w.r.t n, which cannot achieve Oleft(1/nright) when alphain(0,1]. To solve this problem, we propose a variant of gradient perturbation method, max{1,g-Normalized Gradient Perturbation} (m-NGP). We further show that by normalization, the high probability excess population risk bound under assumptions alpha-H{\"o}lder smooth and Polyak-{\L}ojasiewicz condition can achieve Obig(sqrt{p}{nepsilon}big), which is the first Oleft(1/nright) high probability excess population risk bound w.r.t n for differentially private algorithms under non-smooth conditions. Moreover, we evaluate the performance of the new proposed algorithm m-NGP, the experimental results show that m-NGP improves the performance of the differentially private model over real datasets. It demonstrates that m-NGP improves the utility bound and the accuracy of the DP model on real datasets simultaneously.

In this paper, we deal with the gluing of two surfaces, where the gluing locus is assumed to be a curve. We consider a moving frame along the gluing locus, and define developable surfaces with respect to the frame. Considering geometric properties of these developable surfaces, we study the geometry of gluing two surfaces.

Most models in machine learning contain at least one hyperparameter to control for model complexity. Choosing an appropriate set of hyperparameters is both crucial in terms of model accuracy and computationally challenging. In this work we propose an algorithm for the optimization of continuous hyperparameters using inexact gradient information. An advantage of this method is that hyperparameters can be updated before model parameters have fully converged. We also give sufficient conditions for the global convergence of this method, based on regularity conditions of the involved functions and summability of errors. Finally, we validate the empirical performance of this method on the estimation of regularization constants of L2-regularized logistic regression and kernel Ridge regression. Empirical benchmarks indicate that our approach is highly competitive with respect to state of the art methods.

Minibatch optimal transport coupling straightens paths in unconditional flow matching. This leads to computationally less demanding inference as fewer integration steps and less complex numerical solvers can be employed when numerically solving an ordinary differential equation at test time. However, in the conditional setting, minibatch optimal transport falls short. This is because the default optimal transport mapping disregards conditions, resulting in a conditionally skewed prior distribution during training. In contrast, at test time, we have no access to the skewed prior, and instead sample from the full, unbiased prior distribution. This gap between training and testing leads to a subpar performance. To bridge this gap, we propose conditional optimal transport C^2OT that adds a conditional weighting term in the cost matrix when computing the optimal transport assignment. Experiments demonstrate that this simple fix works with both discrete and continuous conditions in 8gaussians-to-moons, CIFAR-10, ImageNet-32x32, and ImageNet-256x256. Our method performs better overall compared to the existing baselines across different function evaluation budgets. Code is available at https://hkchengrex.github.io/C2OT

Recent progress in image generation has sparked research into controlling these models through condition signals, with various methods addressing specific challenges in conditional generation. Instead of proposing another specialized technique, we introduce a simple, unified framework to handle diverse conditional generation tasks involving a specific image-condition correlation. By learning a joint distribution over a correlated image pair (e.g. image and depth) with a diffusion model, our approach enables versatile capabilities via different inference-time sampling schemes, including controllable image generation (e.g. depth to image), estimation (e.g. image to depth), signal guidance, joint generation (image & depth), and coarse control. Previous attempts at unification often introduce significant complexity through multi-stage training, architectural modification, or increased parameter counts. In contrast, our simple formulation requires a single, computationally efficient training stage, maintains the standard model input, and adds minimal learned parameters (15% of the base model). Moreover, our model supports additional capabilities like non-spatially aligned and coarse conditioning. Extensive results show that our single model can produce comparable results with specialized methods and better results than prior unified methods. We also demonstrate that multiple models can be effectively combined for multi-signal conditional generation.

Diffusion inversion is the problem of taking an image and a text prompt that describes it and finding a noise latent that would generate the image. Most current inversion techniques operate by approximately solving an implicit equation and may converge slowly or yield poor reconstructed images. Here, we formulate the problem as finding the roots of an implicit equation and design a method to solve it efficiently. Our solution is based on Newton-Raphson (NR), a well-known technique in numerical analysis. A naive application of NR may be computationally infeasible and tends to converge to incorrect solutions. We describe an efficient regularized formulation that converges quickly to a solution that provides high-quality reconstructions. We also identify a source of inconsistency stemming from prompt conditioning during the inversion process, which significantly degrades the inversion quality. To address this, we introduce a prompt-aware adjustment of the encoding, effectively correcting this issue. Our solution, Regularized Newton-Raphson Inversion, inverts an image within 0.5 sec for latent consistency models, opening the door for interactive image editing. We further demonstrate improved results in image interpolation and generation of rare objects.

New geometric methods for fast evaluation of derivatives of polynomial and rational B\'{e}zier curves are proposed. They apply an algorithm for evaluating polynomial or rational B\'{e}zier curves, which was recently given by the authors. Numerical tests show that the new approach is more efficient than the methods which use the famous de Casteljau algorithm. The algorithms work well even for high-order derivatives of rational B\'{e}zier curves of high degrees.

We present a frame interpolation algorithm that synthesizes multiple intermediate frames from two input images with large in-between motion. Recent methods use multiple networks to estimate optical flow or depth and a separate network dedicated to frame synthesis. This is often complex and requires scarce optical flow or depth ground-truth. In this work, we present a single unified network, distinguished by a multi-scale feature extractor that shares weights at all scales, and is trainable from frames alone. To synthesize crisp and pleasing frames, we propose to optimize our network with the Gram matrix loss that measures the correlation difference between feature maps. Our approach outperforms state-of-the-art methods on the Xiph large motion benchmark. We also achieve higher scores on Vimeo-90K, Middlebury and UCF101, when comparing to methods that use perceptual losses. We study the effect of weight sharing and of training with datasets of increasing motion range. Finally, we demonstrate our model's effectiveness in synthesizing high quality and temporally coherent videos on a challenging near-duplicate photos dataset. Codes and pre-trained models are available at https://film-net.github.io.

In private federated learning (FL), a server aggregates differentially private updates from a large number of clients in order to train a machine learning model. The main challenge in this setting is balancing privacy with both classification accuracy of the learnt model as well as the number of bits communicated between the clients and server. Prior work has achieved a good trade-off by designing a privacy-aware compression mechanism, called the minimum variance unbiased (MVU) mechanism, that numerically solves an optimization problem to determine the parameters of the mechanism. This paper builds upon it by introducing a new interpolation procedure in the numerical design process that allows for a far more efficient privacy analysis. The result is the new Interpolated MVU mechanism that is more scalable, has a better privacy-utility trade-off, and provides SOTA results on communication-efficient private FL on a variety of datasets.

This paper studies the challenging problem of estimating causal effects from observational data, in the presence of unobserved confounders. The two-stage least square (TSLS) method and its variants with a standard instrumental variable (IV) are commonly used to eliminate confounding bias, including the bias caused by unobserved confounders, but they rely on the linearity assumption. Besides, the strict condition of unconfounded instruments posed on a standard IV is too strong to be practical. To address these challenging and practical problems of the standard IV method (linearity assumption and the strict condition), in this paper, we use a conditional IV (CIV) to relax the unconfounded instrument condition of standard IV and propose a non-linear CIV regression with Confounding Balancing Representation Learning, CBRL.CIV, for jointly eliminating the confounding bias from unobserved confounders and balancing the observed confounders, without the linearity assumption. We theoretically demonstrate the soundness of CBRL.CIV. Extensive experiments on synthetic and two real-world datasets show the competitive performance of CBRL.CIV against state-of-the-art IV-based estimators and superiority in dealing with the non-linear situation.

We study the Dirichlet problem for the weighted Schr\"odinger operator \[-\Delta u +Vu = \lambda \rho u,\] where rho is a positive weighting function and V is a potential. Such equations appear naturally in conformal geometry and in the composite membrane problem. Our primary goal is to establish concavity estimates for the principle eigenfunction with respect to conformal connections. Doing so, we obtain new bounds on the fundamental gap problem, which is the difference between the first and second eigenvalues. In particular, we partially resolve a conjecture of Nguyen, Stancu and Wei [IMRN 2022] on the fundamental gap of horoconvex domains. In addition, we obtain a power convexity estimate for solutions to the torsion problem in spherical geometry on convex domains which are not too large.

We study the problem of detecting the correlation between two Gaussian databases XinR^{ntimes d} and Y^{ntimes d}, each composed of n users with d features. This problem is relevant in the analysis of social media, computational biology, etc. We formulate this as a hypothesis testing problem: under the null hypothesis, these two databases are statistically independent. Under the alternative, however, there exists an unknown permutation sigma over the set of n users (or, row permutation), such that X is rho-correlated with Y^sigma, a permuted version of Y. We determine sharp thresholds at which optimal testing exhibits a phase transition, depending on the asymptotic regime of n and d. Specifically, we prove that if rho^2dto0, as dtoinfty, then weak detection (performing slightly better than random guessing) is statistically impossible, irrespectively of the value of n. This compliments the performance of a simple test that thresholds the sum all entries of X^TY. Furthermore, when d is fixed, we prove that strong detection (vanishing error probability) is impossible for any rho<rho^star, where rho^star is an explicit function of d, while weak detection is again impossible as long as rho^2dto0. These results close significant gaps in current recent related studies.

Given an input painting, we reconstruct a time-lapse video of how it may have been painted. We formulate this as an autoregressive image generation problem, in which an initially blank "canvas" is iteratively updated. The model learns from real artists by training on many painting videos. Our approach incorporates text and region understanding to define a set of painting "instructions" and updates the canvas with a novel diffusion-based renderer. The method extrapolates beyond the limited, acrylic style paintings on which it has been trained, showing plausible results for a wide range of artistic styles and genres.

Training deep neural networks for classification often includes minimizing the training loss beyond the zero training error point. In this phase of training, a "neural collapse" behavior has been observed: the variability of features (outputs of the penultimate layer) of within-class samples decreases and the mean features of different classes approach a certain tight frame structure. Recent works analyze this behavior via idealized unconstrained features models where all the minimizers exhibit exact collapse. However, with practical networks and datasets, the features typically do not reach exact collapse, e.g., because deep layers cannot arbitrarily modify intermediate features that are far from being collapsed. In this paper, we propose a richer model that can capture this phenomenon by forcing the features to stay in the vicinity of a predefined features matrix (e.g., intermediate features). We explore the model in the small vicinity case via perturbation analysis and establish results that cannot be obtained by the previously studied models. For example, we prove reduction in the within-class variability of the optimized features compared to the predefined input features (via analyzing gradient flow on the "central-path" with minimal assumptions), analyze the minimizers in the near-collapse regime, and provide insights on the effect of regularization hyperparameters on the closeness to collapse. We support our theory with experiments in practical deep learning settings.

We introduce Cosmos-Transfer, a conditional world generation model that can generate world simulations based on multiple spatial control inputs of various modalities such as segmentation, depth, and edge. In the design, the spatial conditional scheme is adaptive and customizable. It allows weighting different conditional inputs differently at different spatial locations. This enables highly controllable world generation and finds use in various world-to-world transfer use cases, including Sim2Real. We conduct extensive evaluations to analyze the proposed model and demonstrate its applications for Physical AI, including robotics Sim2Real and autonomous vehicle data enrichment. We further demonstrate an inference scaling strategy to achieve real-time world generation with an NVIDIA GB200 NVL72 rack. To help accelerate research development in the field, we open-source our models and code at https://github.com/nvidia-cosmos/cosmos-transfer1.

We present semantic correctness proofs of automatic differentiation (AD). We consider a forward-mode AD method on a higher order language with algebraic data types, and we characterise it as the unique structure preserving macro given a choice of derivatives for basic operations. We describe a rich semantics for differentiable programming, based on diffeological spaces. We show that it interprets our language, and we phrase what it means for the AD method to be correct with respect to this semantics. We show that our characterisation of AD gives rise to an elegant semantic proof of its correctness based on a gluing construction on diffeological spaces. We explain how this is, in essence, a logical relations argument. Throughout, we show how the analysis extends to AD methods for computing higher order derivatives using a Taylor approximation.

Minimax problems are notoriously challenging to optimize. However, we demonstrate that the two-timescale extragradient can be a viable solution. By utilizing dynamical systems theory, we show that it converges to points that satisfy the second-order necessary condition of local minimax points, under a mild condition. This work surpasses all previous results as we eliminate a crucial assumption that the Hessian, with respect to the maximization variable, is nondegenerate.

Data augmentation is becoming essential for improving regression performance in critical applications including manufacturing, climate prediction, and finance. Existing techniques for data augmentation largely focus on classification tasks and do not readily apply to regression tasks. In particular, the recent Mixup techniques for classification have succeeded in improving the model performance, which is reasonable due to the characteristics of the classification task, but has limitations in regression. We show that mixing examples that have large data distances using linear interpolations may have increasingly-negative effects on model performance. Our key idea is thus to limit the distances between examples that are mixed. We propose RegMix, a data augmentation framework for regression that learns for each example how many nearest neighbors it should be mixed with for the best model performance using a validation set. Our experiments conducted both on synthetic and real datasets show that RegMix outperforms state-of-the-art data augmentation baselines applicable to regression.

Multiscale problems are ubiquitous in physics. Numerical simulations of such problems by solving partial differential equations (PDEs) at high resolution are computationally too expensive for many-query scenarios, e.g., uncertainty quantification, remeshing applications, topology optimization, and so forth. This limitation has motivated the application of data-driven surrogate models, where the microscale computations are substituted with a surrogate, usually acting as a black-box mapping between macroscale quantities. These models offer significant speedups but struggle with incorporating microscale physical constraints, such as the balance of linear momentum and constitutive models. In this contribution, we propose Equilibrium Neural Operator (EquiNO) as a complementary physics-informed PDE surrogate for predicting microscale physics and compare it with variational physics-informed neural and operator networks. Our framework, applicable to the so-called multiscale FE^{,2}, computations, introduces the FE-OL approach by integrating the finite element (FE) method with operator learning (OL). We apply the proposed FE-OL approach to quasi-static problems of solid mechanics. The results demonstrate that FE-OL can yield accurate solutions even when confronted with a restricted dataset during model development. Our results show that EquiNO achieves speedup factors exceeding 8000-fold compared to traditional methods and offers an optimal balance between data-driven and physics-based strategies.

We present Scalable Interpolant Transformers (SiT), a family of generative models built on the backbone of Diffusion Transformers (DiT). The interpolant framework, which allows for connecting two distributions in a more flexible way than standard diffusion models, makes possible a modular study of various design choices impacting generative models built on dynamical transport: using discrete vs. continuous time learning, deciding the objective for the model to learn, choosing the interpolant connecting the distributions, and deploying a deterministic or stochastic sampler. By carefully introducing the above ingredients, SiT surpasses DiT uniformly across model sizes on the conditional ImageNet 256x256 benchmark using the exact same backbone, number of parameters, and GFLOPs. By exploring various diffusion coefficients, which can be tuned separately from learning, SiT achieves an FID-50K score of 2.06.

Time series data from observations of black hole ringdown gravitational waves are often analyzed in the time domain by using damped sinusoid models with acyclic boundary conditions. Data conditioning operations, including downsampling, filtering, and the choice of data segment duration, reduce the computational cost of such analyses and can improve numerical stability. Here we analyze simulated damped sinsuoid signals to illustrate how data conditioning operations, if not carefully applied, can undesirably alter the analysis' posterior distributions. We discuss how currently implemented downsampling and filtering methods, if applied too aggressively, can introduce systematic errors and skew tests of general relativity. These issues arise because current downsampling and filtering methods do not operate identically on the data and model. Alternative downsampling and filtering methods which identically operate on the data and model may be achievable, but we argue that the current operations can still be implemented safely. We also show that our preferred anti-alias filtering technique, which has an instantaneous frequency-domain response at its roll-off frequency, preserves the structure of posterior distributions better than other commonly used filters with transient frequency-domain responses. Lastly, we highlight that exceptionally long data segments may need to be analyzed in cases where thin lines in the noise power spectral density overlap with central signal frequencies. Our findings may be broadly applicable to any analysis of truncated time domain data with acyclic boundary conditions.

We propose ScaledGD(\lambda), a preconditioned gradient descent method to tackle the low-rank matrix sensing problem when the true rank is unknown, and when the matrix is possibly ill-conditioned. Using overparametrized factor representations, ScaledGD(\lambda) starts from a small random initialization, and proceeds by gradient descent with a specific form of damped preconditioning to combat bad curvatures induced by overparameterization and ill-conditioning. At the expense of light computational overhead incurred by preconditioners, ScaledGD(\lambda) is remarkably robust to ill-conditioning compared to vanilla gradient descent (GD) even with overprameterization. Specifically, we show that, under the Gaussian design, ScaledGD(\lambda) converges to the true low-rank matrix at a constant linear rate after a small number of iterations that scales only logarithmically with respect to the condition number and the problem dimension. This significantly improves over the convergence rate of vanilla GD which suffers from a polynomial dependency on the condition number. Our work provides evidence on the power of preconditioning in accelerating the convergence without hurting generalization in overparameterized learning.

Image-based virtual try-on aims to synthesize an image of a person wearing a given clothing item. To solve the task, the existing methods warp the clothing item to fit the person's body and generate the segmentation map of the person wearing the item before fusing the item with the person. However, when the warping and the segmentation generation stages operate individually without information exchange, the misalignment between the warped clothes and the segmentation map occurs, which leads to the artifacts in the final image. The information disconnection also causes excessive warping near the clothing regions occluded by the body parts, so-called pixel-squeezing artifacts. To settle the issues, we propose a novel try-on condition generator as a unified module of the two stages (i.e., warping and segmentation generation stages). A newly proposed feature fusion block in the condition generator implements the information exchange, and the condition generator does not create any misalignment or pixel-squeezing artifacts. We also introduce discriminator rejection that filters out the incorrect segmentation map predictions and assures the performance of virtual try-on frameworks. Experiments on a high-resolution dataset demonstrate that our model successfully handles the misalignment and occlusion, and significantly outperforms the baselines. Code is available at https://github.com/sangyun884/HR-VITON.

Controllable layout generation refers to the process of creating a plausible visual arrangement of elements within a graphic design (e.g., document and web designs) with constraints representing design intentions. Although recent diffusion-based models have achieved state-of-the-art FID scores, they tend to exhibit more pronounced misalignment compared to earlier transformer-based models. In this work, we propose the LAyout Constraint diffusion modEl (LACE), a unified model to handle a broad range of layout generation tasks, such as arranging elements with specified attributes and refining or completing a coarse layout design. The model is based on continuous diffusion models. Compared with existing methods that use discrete diffusion models, continuous state-space design can enable the incorporation of differentiable aesthetic constraint functions in training. For conditional generation, we introduce conditions via masked input. Extensive experiment results show that LACE produces high-quality layouts and outperforms existing state-of-the-art baselines.

Diffusion models have demonstrated impressive abilities in generating photo-realistic and creative images. To offer more controllability for the generation process, existing studies, termed as early-constraint methods in this paper, leverage extra conditions and incorporate them into pre-trained diffusion models. Particularly, some of them adopt condition-specific modules to handle conditions separately, where they struggle to generalize across other conditions. Although follow-up studies present unified solutions to solve the generalization problem, they also require extra resources to implement, e.g., additional inputs or parameter optimization, where more flexible and efficient solutions are expected to perform steerable guided image synthesis. In this paper, we present an alternative paradigm, namely Late-Constraint Diffusion (LaCon), to simultaneously integrate various conditions into pre-trained diffusion models. Specifically, LaCon establishes an alignment between the external condition and the internal features of diffusion models, and utilizes the alignment to incorporate the target condition, guiding the sampling process to produce tailored results. Experimental results on COCO dataset illustrate the effectiveness and superior generalization capability of LaCon under various conditions and settings. Ablation studies investigate the functionalities of different components in LaCon, and illustrate its great potential to serve as an efficient solution to offer flexible controllability for diffusion models.

The premise of identifiable and causal representation learning is to improve the current representation learning paradigm in terms of generalizability or robustness. Despite recent progress in questions of identifiability, more theoretical results demonstrating concrete advantages of these methods for downstream tasks are needed. In this paper, we consider the task of intervention extrapolation: predicting how interventions affect an outcome, even when those interventions are not observed at training time, and show that identifiable representations can provide an effective solution to this task even if the interventions affect the outcome non-linearly. Our setup includes an outcome Y, observed features X, which are generated as a non-linear transformation of latent features Z, and exogenous action variables A, which influence Z. The objective of intervention extrapolation is to predict how interventions on A that lie outside the training support of A affect Y. Here, extrapolation becomes possible if the effect of A on Z is linear and the residual when regressing Z on A has full support. As Z is latent, we combine the task of intervention extrapolation with identifiable representation learning, which we call Rep4Ex: we aim to map the observed features X into a subspace that allows for non-linear extrapolation in A. We show that the hidden representation is identifiable up to an affine transformation in Z-space, which is sufficient for intervention extrapolation. The identifiability is characterized by a novel constraint describing the linearity assumption of A on Z. Based on this insight, we propose a method that enforces the linear invariance constraint and can be combined with any type of autoencoder. We validate our theoretical findings through synthetic experiments and show that our approach succeeds in predicting the effects of unseen interventions.

A Milstein-type method is proposed for some highly non-linear non-autonomous time-changed stochastic differential equations (SDEs). The spatial variables in the coefficients of the time-changed SDEs satisfy the super-linear growth condition and the temporal variables obey some H\"older's continuity condition. The strong convergence in the finite time is studied and the convergence order is obtained.

Automatic analysis of the enormous sets of images is a critical task in life sciences. This faces many challenges such as: algorithms are highly parameterized, significant human input is intertwined, and lacking a standard meta-visualization approach. This paper proposes an alternative iterative approach for optimizing input parameters, saving time by minimizing the user involvement, and allowing for understanding the workflow of algorithms and discovering new ones. The main focus is on developing an interactive visualization technique that enables users to analyze the relationships between sampled input parameters and corresponding output. This technique is implemented as a prototype called Veni Vidi Vici, or "I came, I saw, I conquered." This strategy is inspired by the mathematical formulas of numbering computable functions and is developed atop ImageJ, a scientific image processing program. A case study is presented to investigate the proposed framework. Finally, the paper explores some potential future issues in the application of the proposed approach in parameter space analysis in visualization.

In the misspecified kernel ridge regression problem, researchers usually assume the underground true function f_{rho}^{*} in [H]^{s}, a less-smooth interpolation space of a reproducing kernel Hilbert space (RKHS) H for some sin (0,1). The existing minimax optimal results require |f_{rho}^{*}|_{L^{infty}}<infty which implicitly requires s > alpha_{0} where alpha_{0}in (0,1) is the embedding index, a constant depending on H. Whether the KRR is optimal for all sin (0,1) is an outstanding problem lasting for years. In this paper, we show that KRR is minimax optimal for any sin (0,1) when the H is a Sobolev RKHS.

We propose a new approach for generative modeling based on training a neural network to be idempotent. An idempotent operator is one that can be applied sequentially without changing the result beyond the initial application, namely f(f(z))=f(z). The proposed model f is trained to map a source distribution (e.g, Gaussian noise) to a target distribution (e.g. realistic images) using the following objectives: (1) Instances from the target distribution should map to themselves, namely f(x)=x. We define the target manifold as the set of all instances that f maps to themselves. (2) Instances that form the source distribution should map onto the defined target manifold. This is achieved by optimizing the idempotence term, f(f(z))=f(z) which encourages the range of f(z) to be on the target manifold. Under ideal assumptions such a process provably converges to the target distribution. This strategy results in a model capable of generating an output in one step, maintaining a consistent latent space, while also allowing sequential applications for refinement. Additionally, we find that by processing inputs from both target and source distributions, the model adeptly projects corrupted or modified data back to the target manifold. This work is a first step towards a ``global projector'' that enables projecting any input into a target data distribution.

Let E be the HNN-extension of a group B with subgroups H and K associated according to an isomorphism varphicolon H to K. Suppose that H and K are normal in B and (H cap K)varphi = H cap K. Under these assumptions, we prove necessary and sufficient conditions for E to be residually a C-group, where C is a class of groups closed under taking subgroups, quotient groups, and unrestricted wreath products. Among other things, these conditions give new facts on the residual finiteness and the residual p-finiteness of the group E.

Recent advancements in differentiable rendering and 3D reasoning have driven exciting results in novel view synthesis from a single image. Despite realistic results, methods are limited to relatively small view change. In order to synthesize immersive scenes, models must also be able to extrapolate. We present an approach that fuses 3D reasoning with autoregressive modeling to outpaint large view changes in a 3D-consistent manner, enabling scene synthesis. We demonstrate considerable improvement in single image large-angle view synthesis results compared to a variety of methods and possible variants across simulated and real datasets. In addition, we show increased 3D consistency compared to alternative accumulation methods. Project website: https://crockwell.github.io/pixelsynth/

With the exponential growth of video traffic, traditional video streaming systems are approaching their limits in compression efficiency and communication capacity. To further reduce bitrate while maintaining quality, we propose Promptus, a disruptive semantic communication system that streaming prompts instead of video content, which represents real-world video frames with a series of "prompts" for delivery and employs Stable Diffusion to generate videos at the receiver. To ensure that the generated video is pixel-aligned with the original video, a gradient descent-based prompt fitting framework is proposed. Further, a low-rank decomposition-based bitrate control algorithm is introduced to achieve adaptive bitrate. For inter-frame compression, an interpolation-aware fitting algorithm is proposed. Evaluations across various video genres demonstrate that, compared to H.265, Promptus can achieve more than a 4x bandwidth reduction while preserving the same perceptual quality. On the other hand, at extremely low bitrates, Promptus can enhance the perceptual quality by 0.139 and 0.118 (in LPIPS) compared to VAE and H.265, respectively, and decreases the ratio of severely distorted frames by 89.3% and 91.7%. Our work opens up a new paradigm for efficient video communication. Promptus is open-sourced at: https://github.com/JiangkaiWu/Promptus.

We consider the inverse Stefan type free boundary problem, where information on the boundary heat flux and density of the sources are missing and must be found along with the temperature and the free boundary. We pursue optimal control framework where boundary heat flux, density of sources, and free boundary are components of the control vector. The optimality criteria consists of the minimization of the L_2-norm declinations of the temperature measurements at the final moment, phase transition temperature, and final position of the free boundary. We prove the Frechet differentiability in Besov spaces, and derive the formula for the Frechet differential under minimal regularity assumptions on the data. The result implies a necessary condition for optimal control and opens the way to the application of projective gradient methods in Besov spaces for the numerical solution of the inverse Stefan problem.

In this paper, we examine the solvability of a functional equation in a Lipschitz space. As an application, we use our result to determine the existence and uniqueness of solutions to an equation describing a specific type of choice behavior model for the learning process of the paradise fish. Finally, we present some concrete examples where, using numerical techniques, we obtain approximations to the solution of the functional equation. As the straightforward Picard's iteration can be very expensive, we show that an analytical suboptimal least-squares approximation can be chosen in practice, resulting in very good accuracy.

We derive a minimalist but powerful deterministic denoising-diffusion model. While denoising diffusion has shown great success in many domains, its underlying theory remains largely inaccessible to non-expert users. Indeed, an understanding of graduate-level concepts such as Langevin dynamics or score matching appears to be required to grasp how it works. We propose an alternative approach that requires no more than undergrad calculus and probability. We consider two densities and observe what happens when random samples from these densities are blended (linearly interpolated). We show that iteratively blending and deblending samples produces random paths between the two densities that converge toward a deterministic mapping. This mapping can be evaluated with a neural network trained to deblend samples. We obtain a model that behaves like deterministic denoising diffusion: it iteratively maps samples from one density (e.g., Gaussian noise) to another (e.g., cat images). However, compared to the state-of-the-art alternative, our model is simpler to derive, simpler to implement, more numerically stable, achieves higher quality results in our experiments, and has interesting connections to computer graphics.

We learn a latent space for easy capture, consistent interpolation, and efficient reproduction of visual material appearance. When users provide a photo of a stationary natural material captured under flashlight illumination, first it is converted into a latent material code. Then, in the second step, conditioned on the material code, our method produces an infinite and diverse spatial field of BRDF model parameters (diffuse albedo, normals, roughness, specular albedo) that subsequently allows rendering in complex scenes and illuminations, matching the appearance of the input photograph. Technically, we jointly embed all flash images into a latent space using a convolutional encoder, and -- conditioned on these latent codes -- convert random spatial fields into fields of BRDF parameters using a convolutional neural network (CNN). We condition these BRDF parameters to match the visual characteristics (statistics and spectra of visual features) of the input under matching light. A user study compares our approach favorably to previous work, even those with access to BRDF supervision.

Physics-informed deep learning often faces optimization challenges due to the complexity of solving partial differential equations (PDEs), which involve exploring large solution spaces, require numerous iterations, and can lead to unstable training. These challenges arise particularly from the ill-conditioning of the optimization problem caused by the differential terms in the loss function. To address these issues, we propose learning a solver, i.e., solving PDEs using a physics-informed iterative algorithm trained on data. Our method learns to condition a gradient descent algorithm that automatically adapts to each PDE instance, significantly accelerating and stabilizing the optimization process and enabling faster convergence of physics-aware models. Furthermore, while traditional physics-informed methods solve for a single PDE instance, our approach extends to parametric PDEs. Specifically, we integrate the physical loss gradient with PDE parameters, allowing our method to solve over a distribution of PDE parameters, including coefficients, initial conditions, and boundary conditions. We demonstrate the effectiveness of our approach through empirical experiments on multiple datasets, comparing both training and test-time optimization performance. The code is available at https://github.com/2ailesB/neural-parametric-solver.

In this article, we continue the development of the Riemann-Hilbert formalism for studying the asymptotics of Toeplitz+Hankel determinants with non-identical symbols, which we initiated in GI. In GI, we showed that the Riemann-Hilbert problem we formulated admits the Deift-Zhou nonlinear steepest descent analysis, but with a special restriction on the winding numbers of the associated symbols. In particular, the most natural case, namely zero winding numbers, is not allowed. A principal goal of this paper is to develop a framework that extends the asymptotic analysis of Toeplitz+Hankel determinants to a broader range of winding-number configurations. As an application, we consider the case in which the winding numbers of the Szego-type Toeplitz and Hankel symbols are zero and one, respectively, and compute the asymptotics of the norms of the corresponding system of orthogonal polynomials.

In training neural networks, it is common practice to use partial gradients computed over batches, mostly very small subsets of the training set. This approach is motivated by the argument that such a partial gradient is close to the true one, with precision growing only with the square root of the batch size. A theoretical justification is with the help of stochastic approximation theory. However, the conditions for the validity of this theory are not satisfied in the usual learning rate schedules. Batch processing is also difficult to combine with efficient second-order optimization methods. This proposal is based on another hypothesis: the loss minimum of the training set can be expected to be well-approximated by the minima of its subsets. Such subset minima can be computed in a fraction of the time necessary for optimizing over the whole training set. This hypothesis has been tested with the help of the MNIST, CIFAR-10, and CIFAR-100 image classification benchmarks, optionally extended by training data augmentation. The experiments have confirmed that results equivalent to conventional training can be reached. In summary, even small subsets are representative if the overdetermination ratio for the given model parameter set sufficiently exceeds unity. The computing expense can be reduced to a tenth or less.

We present semantic correctness proofs of Automatic Differentiation (AD). We consider a forward-mode AD method on a higher order language with algebraic data types, and we characterise it as the unique structure preserving macro given a choice of derivatives for basic operations. We describe a rich semantics for differentiable programming, based on diffeological spaces. We show that it interprets our language, and we phrase what it means for the AD method to be correct with respect to this semantics. We show that our characterisation of AD gives rise to an elegant semantic proof of its correctness based on a gluing construction on diffeological spaces. We explain how this is, in essence, a logical relations argument. Finally, we sketch how the analysis extends to other AD methods by considering a continuation-based method.

Mixup is a powerful data augmentation method that interpolates between two or more examples in the input or feature space and between the corresponding target labels. Many recent mixup methods focus on cutting and pasting two or more objects into one image, which is more about efficient processing than interpolation. However, how to best interpolate images is not well defined. In this sense, mixup has been connected to autoencoders, because often autoencoders "interpolate well", for instance generating an image that continuously deforms into another. In this work, we revisit mixup from the interpolation perspective and introduce AlignMix, where we geometrically align two images in the feature space. The correspondences allow us to interpolate between two sets of features, while keeping the locations of one set. Interestingly, this gives rise to a situation where mixup retains mostly the geometry or pose of one image and the texture of the other, connecting it to style transfer. More than that, we show that an autoencoder can still improve representation learning under mixup, without the classifier ever seeing decoded images. AlignMix outperforms state-of-the-art mixup methods on five different benchmarks.

Diffusion models have exhibited exciting capabilities in generating images and are also very promising for video creation. However, the inference speed of diffusion models is limited by the slow sampling process, restricting its use cases. The sequential denoising steps required for generating a single sample could take tens or hundreds of iterations and thus have become a significant bottleneck. This limitation is more salient for applications that are interactive in nature or require small latency. To address this challenge, we propose Partially Conditioned Patch Parallelism (PCPP) to accelerate the inference of high-resolution diffusion models. Using the fact that the difference between the images in adjacent diffusion steps is nearly zero, Patch Parallelism (PP) leverages multiple GPUs communicating asynchronously to compute patches of an image in multiple computing devices based on the entire image (all patches) in the previous diffusion step. PCPP develops PP to reduce computation in inference by conditioning only on parts of the neighboring patches in each diffusion step, which also decreases communication among computing devices. As a result, PCPP decreases the communication cost by around 70% compared to DistriFusion (the state of the art implementation of PP) and achieves 2.36sim 8.02times inference speed-up using 4sim 8 GPUs compared to 2.32sim 6.71times achieved by DistriFusion depending on the computing device configuration and resolution of generation at the cost of a possible decrease in image quality. PCPP demonstrates the potential to strike a favorable trade-off, enabling high-quality image generation with substantially reduced latency.

This article establishes a complete approximate axiomatization for the real-closed field R expanded with all differentially-defined functions, including special functions such as sin(x), cos(x), e^x, dots. Every true sentence is provable up to some numerical approximation, and the truth of such approximations converge under mild conditions. Such an axiomatization is a fragment of the axiomatization for differential dynamic logic, and is therefore a finite extension of the axiomatization of real-closed fields. Furthermore, the numerical approximations approximate formulas containing special function symbols by FOL_{R} formulas, improving upon earlier decidability results only concerning closed sentences.

Offline optimization is an important task in numerous material engineering domains where online experimentation to collect data is too expensive and needs to be replaced by an in silico maximization of a surrogate of the black-box function. Although such a surrogate can be learned from offline data, its prediction might not be reliable outside the offline data regime, which happens when the surrogate has narrow prediction margin and is (therefore) sensitive to small perturbations of its parameterization. This raises the following questions: (1) how to regulate the sensitivity of a surrogate model; and (2) whether conditioning an offline optimizer with such less sensitive surrogate will lead to better optimization performance. To address these questions, we develop an optimizable sensitivity measurement for the surrogate model, which then inspires a sensitivity-informed regularizer that is applicable to a wide range of offline optimizers. This development is both orthogonal and synergistic to prior research on offline optimization, which is demonstrated in our extensive experiment benchmark.

We introduce a general, flexible, parametric survival modelling framework which encompasses key shapes of hazard function (constant, increasing, decreasing, up-then-down, down-then-up), various common survival distributions (log-logistic, Burr type XII, Weibull, Gompertz), and includes defective distributions (i.e., cure models). This generality is achieved using four basic distributional parameters: two scale-type parameters and two shape parameters. Generalising to covariate dependence, the scale-type regression components correspond to accelerated failure time (AFT) and proportional hazards (PH) models. Therefore, this general formulation unifies the most popular survival models which allows us to consider the practical value of possible modelling choices for survival data. Furthermore, in line with our proposed flexible baseline distribution, we advocate the use of multi-parameter regression in which more than one distributional parameter depends on covariates - rather than the usual convention of having a single covariate-dependent (scale) parameter. While many choices are available, we suggest introducing covariates through just one or other of the two scale parameters, which covers AFT and PH models, in combination with a `power' shape parameter, which allows for more complex non-AFT/non-PH effects, while the other shape parameter remains covariate-independent, and handles automatic selection of the baseline distribution. We explore inferential issues in simulations, both with and without a covariate, with particular focus on evidence concerning the need, or otherwise, to include both AFT and PH parameters. We illustrate the efficacy of our modelling framework by investigating differences between treatment groups using data from a lung cancer study and a melanoma study. Censoring is accommodated throughout.

We consider a fair division model in which agents have positive, zero and negative utilities for items. For this model, we analyse one existing fairness property - EFX - and three new and related properties - EFX_0, EFX^3 and EF1^3 - in combination with Pareto-optimality. With general utilities, we give a modified version of an existing algorithm for computing an EF1^3 allocation. With -alpha/0/alpha utilities, this algorithm returns an EFX^3 and PO allocation. With absolute identical utilities, we give a new algorithm for an EFX and PO allocation. With -alpha/0/beta utilities, this algorithm also returns such an allocation. We report some new impossibility results as well.

The proposed method aims to approximate a solution of a fluid-fluid interaction problem in case of low viscosities. The nonlinear interface condition on the joint boundary allows for this problem to be viewed as a simplified version of the atmosphere-ocean coupling. Thus, the proposed method should be viewed as potentially applicable to air-sea coupled flows in turbulent regime. The method consists of two key ingredients. The geometric averaging approach is used for efficient and stable decoupling of the problem, which would allow for the usage of preexisting codes for the air and sea domain separately, as "black boxes". This is combined with the variational multiscale stabilization technique for treating flows at high Reynolds numbers. We prove the stability and accuracy of the method and provide several numerical tests to assess both the quantitative and qualitative features of the computed solution.

Instrumental variable (IV) methods are used to estimate causal effects in settings with unobserved confounding, where we cannot directly experiment on the treatment variable. Instruments are variables which only affect the outcome indirectly via the treatment variable(s). Most IV applications focus on low-dimensional treatments and crucially require at least as many instruments as treatments. This assumption is restrictive: in the natural sciences we often seek to infer causal effects of high-dimensional treatments (e.g., the effect of gene expressions or microbiota on health and disease), but can only run few experiments with a limited number of instruments (e.g., drugs or antibiotics). In such underspecified problems, the full treatment effect is not identifiable in a single experiment even in the linear case. We show that one can still reliably recover the projection of the treatment effect onto the instrumented subspace and develop techniques to consistently combine such partial estimates from different sets of instruments. We then leverage our combined estimators in an algorithm that iteratively proposes the most informative instruments at each round of experimentation to maximize the overall information about the full causal effect.

This research enhances linear regression models by integrating a Kalman filter and analysing curve areas to minimize loss. The goal is to develop an optimal linear regression equation using stochastic gradient descent (SGD) for weight updating. Our approach involves a stepwise process, starting with user-defined parameters. The linear regression model is trained using SGD, tracking weights and loss separately and zipping them finally. A Kalman filter is then trained based on weight and loss arrays to predict the next consolidated weights. Predictions result from multiplying input averages with weights, evaluated for loss to form a weight-versus-loss curve. The curve's equation is derived using the two-point formula, and area under the curve is calculated via integration. The linear regression equation with minimum area becomes the optimal curve for prediction. Benefits include avoiding constant weight updates via gradient descent and working with partial datasets, unlike methods needing the entire set. However, computational complexity should be considered. The Kalman filter's accuracy might diminish beyond a certain prediction range.

Many problems in astrophysics cover multiple orders of magnitude in spatial and temporal scales. While simulating systems that experience rapid changes in these conditions, it is essential to adapt the (time-) step size to capture the behavior of the system during those rapid changes and use a less accurate time step at other, less demanding, moments. We encounter three problems with traditional methods. Firstly, making such changes requires expert knowledge of the astrophysics as well as of the details of the numerical implementation. Secondly, some parameters that determine the time-step size are fixed throughout the simulation, which means that they do not adapt to the rapidly changing conditions of the problem. Lastly, we would like the choice of time-step size to balance accuracy and computation effort. We address these challenges with Reinforcement Learning by training it to select the time-step size dynamically. We use the integration of a system of three equal-mass bodies that move due to their mutual gravity as an example of its application. With our method, the selected integration parameter adapts to the specific requirements of the problem, both in terms of computation time and accuracy while eliminating the expert knowledge needed to set up these simulations. Our method produces results competitive to existing methods and improve the results found with the most commonly-used values of time-step parameter. This method can be applied to other integrators without further retraining. We show that this extrapolation works for variable time-step integrators but does not perform to the desired accuracy for fixed time-step integrators.

In this paper, we study the existence and uniqueness of solutions to the Euler equations with initial conditions that exhibit analytic regularity near the boundary and Sobolev regularity away from it. A key contribution of this work is the introduction of the diamond-analyticity framework, which captures the spatial decay of the analyticity radius in a structured manner, improving upon uniform analyticity approaches. We employ the Leray projection and a nonstandard mollification technique to demonstrate that the quotient between the imaginary and real parts of the analyticity radius remains unrestricted, thus extending the analyticity persistence results beyond traditional constraints. Our methodology combines analytic-Sobolev estimates with an iterative scheme which is nonstandard in the Cauchy-Kowalevskaya framework, ensuring rigorous control over the evolution of the solution. These results contribute to a deeper understanding of the interplay between analyticity and boundary effects in fluid equations. They might have implications for the study of the inviscid limit of the Navier-Stokes equations and the role of complex singularities in fluid dynamics.

Long video generation has gained increasing attention due to its widespread applications in fields such as entertainment and simulation. Despite advances, synthesizing temporally coherent and visually compelling long sequences remains a formidable challenge. Conventional approaches often synthesize long videos by sequentially generating and concatenating short clips, or generating key frames and then interpolate the intermediate frames in a hierarchical manner. However, both of them still remain significant challenges, leading to issues such as temporal repetition or unnatural transitions. In this paper, we revisit the hierarchical long video generation pipeline and introduce LumosFlow, a framework introduce motion guidance explicitly. Specifically, we first employ the Large Motion Text-to-Video Diffusion Model (LMTV-DM) to generate key frames with larger motion intervals, thereby ensuring content diversity in the generated long videos. Given the complexity of interpolating contextual transitions between key frames, we further decompose the intermediate frame interpolation into motion generation and post-hoc refinement. For each pair of key frames, the Latent Optical Flow Diffusion Model (LOF-DM) synthesizes complex and large-motion optical flows, while MotionControlNet subsequently refines the warped results to enhance quality and guide intermediate frame generation. Compared with traditional video frame interpolation, we achieve 15x interpolation, ensuring reasonable and continuous motion between adjacent frames. Experiments show that our method can generate long videos with consistent motion and appearance. Code and models will be made publicly available upon acceptance. Our project page: https://jiahaochen1.github.io/LumosFlow/

We show how to transform a non-differentiable rasterizer into a differentiable one with minimal engineering efforts and no external dependencies (no Pytorch/Tensorflow). We rely on Stochastic Gradient Estimation, a technique that consists of rasterizing after randomly perturbing the scene's parameters such that their gradient can be stochastically estimated and descended. This method is simple and robust but does not scale in dimensionality (number of scene parameters). Our insight is that the number of parameters contributing to a given rasterized pixel is bounded. Estimating and averaging gradients on a per-pixel basis hence bounds the dimensionality of the underlying optimization problem and makes the method scalable. Furthermore, it is simple to track per-pixel contributing parameters by rasterizing ID- and UV-buffers, which are trivial additions to a rasterization engine if not already available. With these minor modifications, we obtain an in-engine optimizer for 3D assets with millions of geometry and texture parameters.

We prove a sharp Hardy inequality for fractional integrals for functions that are supported on a general domain. The constant is the same as the one for the half-space and hence our result settles a recent conjecture of Bogdan and Dyda.

There is a growing gap between the impressive results of deep image generative models and classical algorithms that offer theoretical guarantees. The former suffer from mode collapse or memorization issues, limiting their application to scientific data. The latter require restrictive assumptions such as log-concavity to escape the curse of dimensionality. We partially bridge this gap by introducing conditionally strongly log-concave (CSLC) models, which factorize the data distribution into a product of conditional probability distributions that are strongly log-concave. This factorization is obtained with orthogonal projectors adapted to the data distribution. It leads to efficient parameter estimation and sampling algorithms, with theoretical guarantees, although the data distribution is not globally log-concave. We show that several challenging multiscale processes are conditionally log-concave using wavelet packet orthogonal projectors. Numerical results are shown for physical fields such as the varphi^4 model and weak lensing convergence maps with higher resolution than in previous works.

Synthesizing optimal controllers for dynamical systems often involves solving optimization problems with hard real-time constraints. These constraints determine the class of numerical methods that can be applied: computationally expensive but accurate numerical routines are replaced by fast and inaccurate methods, trading inference time for solution accuracy. This paper provides techniques to improve the quality of optimized control policies given a fixed computational budget. We achieve the above via a hypersolvers approach, which hybridizes a differential equation solver and a neural network. The performance is evaluated in direct and receding-horizon optimal control tasks in both low and high dimensions, where the proposed approach shows consistent Pareto improvements in solution accuracy and control performance.

Data used for analytics and machine learning often take the form of tables with categorical entries. We introduce a family of lossless compression algorithms for such data that proceed in four steps: (i) Estimate latent variables associated to rows and columns; (ii) Partition the table in blocks according to the row/column latents; (iii) Apply a sequential (e.g. Lempel-Ziv) coder to each of the blocks; (iv) Append a compressed encoding of the latents. We evaluate it on several benchmark datasets, and study optimal compression in a probabilistic model for that tabular data, whereby latent values are independent and table entries are conditionally independent given the latent values. We prove that the model has a well defined entropy rate and satisfies an asymptotic equipartition property. We also prove that classical compression schemes such as Lempel-Ziv and finite-state encoders do not achieve this rate. On the other hand, the latent estimation strategy outlined above achieves the optimal rate.

We introduced a novel method to solve Bayesian inverse problems governed by PDE equations with a hybrid two-level MCMC where we took advantage of the AI surrogate model speed and the accuracy of numerical models. We show theoretically the potential to solve Bayesian inverse problems accurately with only a small number of numerical samples when the AI surrogate model error is small. Several numerical experiment results are included which demonstrates the advantage of the hybrid method.

We present a new algorithm to optimize distributions defined implicitly by parameterized stochastic diffusions. Doing so allows us to modify the outcome distribution of sampling processes by optimizing over their parameters. We introduce a general framework for first-order optimization of these processes, that performs jointly, in a single loop, optimization and sampling steps. This approach is inspired by recent advances in bilevel optimization and automatic implicit differentiation, leveraging the point of view of sampling as optimization over the space of probability distributions. We provide theoretical guarantees on the performance of our method, as well as experimental results demonstrating its effectiveness in real-world settings.

The objective is the design of a Cellular Automata rule that can form patterns with 'touching' loops. A loop is defined as a closed path of 1-cells in a 2D grid on a zero background and with a zero border. A path cell is connected with two of its adjacent neighbors. In touching loops a path cell is also allowed to touch another on a diagonal. A CA rule was designed that can evolve stable touching loop patterns. The rule tries to cover the 2D space by overlapping tiles. The rule uses so-called templates, 5 x 5 matching patterns which are systematically derived from the given set of 3 x 3 tiles. The rule checks the pattern being evolved against a list of templates. If the outer neighbors of a template match, then the cell's state is set to the template's center value. Noise is injected if there is no matching template, or the tiles are not properly assembled. Thereby the evolution is driven to the desired loop patterns.

Point-interactive image colorization aims to colorize grayscale images when a user provides the colors for specific locations. It is essential for point-interactive colorization methods to appropriately propagate user-provided colors (i.e., user hints) in the entire image to obtain a reasonably colorized image with minimal user effort. However, existing approaches often produce partially colorized results due to the inefficient design of stacking convolutional layers to propagate hints to distant relevant regions. To address this problem, we present iColoriT, a novel point-interactive colorization Vision Transformer capable of propagating user hints to relevant regions, leveraging the global receptive field of Transformers. The self-attention mechanism of Transformers enables iColoriT to selectively colorize relevant regions with only a few local hints. Our approach colorizes images in real-time by utilizing pixel shuffling, an efficient upsampling technique that replaces the decoder architecture. Also, in order to mitigate the artifacts caused by pixel shuffling with large upsampling ratios, we present the local stabilizing layer. Extensive quantitative and qualitative results demonstrate that our approach highly outperforms existing methods for point-interactive colorization, producing accurately colorized images with a user's minimal effort. Official codes are available at https://pmh9960.github.io/research/iColoriT

We present an open-source database of superconducting quantum device designs that may be used as the starting point for customized devices. Each design can be generated programmatically using the open-source Qiskit Metal package, and simulated using finite-element electromagnetic solvers. We present a robust workflow for achieving high accuracy on design simulations. Many designs in the database are experimentally validated, showing excellent agreement between simulated and measured parameters. Our database includes a front-end interface that allows users to generate ``best-guess'' designs based on desired circuit parameters. This project lowers the barrier to entry for research groups seeking to make a new class of devices by providing them a well-characterized starting point from which to refine their designs.

For the evolutionary Stokes problem with dynamic boundary condition we show maximal regularity of weak solutions in time. Due to the characteriation of R-sectorial operators on Hilbert spaces, the proof reduces to finding the correct functional analytic setting and proving that an operator is sectorial, i.e. generates an analytic semigroup.

In an industrial group like Safran, numerical simulations of physical phenomena are integral to most design processes. At Safran's corporate research center, we enhance these processes by developing fast and reliable surrogate models for various physics. We focus here on two technologies developed in recent years. The first is a physical reduced-order modeling method for non-linear structural mechanics and thermal analysis, used for calculating the lifespan of high-pressure turbine blades and performing heat analysis of high-pressure compressors. The second technology involves learning physics simulations with non-parameterized geometrical variability using classical machine learning tools, such as Gaussian process regression. Finally, we present our contributions to the open-source and open-data community.

The Shadowing Principle of Manin has proved a valuable tool for addressing questions of quantitative topology raised by Gromov in the late 1900s. The principle informally provides a way for bounded algebraic maps between differential graded algebras to be translated into nearby genuine maps between their geometric realizations. We extend this principle to finite towers of principal K(G,n) fibrations, and in particular apply this construction to nilpotent spaces. As a specific application of the extended principle, we provide upper bounds on the asymptotic behavior of volumes of nullhomotopies of Lipschitz maps into nilpotent spaces. We further refine these bounds in the case when c = 1 to nearly meet those of the simply connected setting. We similarly refine these bounds in the event the target space is coformal, and demonstrate that the bounds in this setting are nearly sharp.

Reconstructing dynamic assets from video data is central to many in computer vision and graphics tasks. Existing 4D reconstruction approaches are limited by category-specific models or slow optimization-based methods. Inspired by the recent Large Reconstruction Model (LRM), we present the Large Interpolation Model (LIM), a transformer-based feed-forward solution, guided by a novel causal consistency loss, for interpolating implicit 3D representations across time. Given implicit 3D representations at times t_0 and t_1, LIM produces a deformed shape at any continuous time tin[t_0,t_1], delivering high-quality interpolated frames in seconds. Furthermore, LIM allows explicit mesh tracking across time, producing a consistently uv-textured mesh sequence ready for integration into existing production pipelines. We also use LIM, in conjunction with a diffusion-based multiview generator, to produce dynamic 4D reconstructions from monocular videos. We evaluate LIM on various dynamic datasets, benchmarking against image-space interpolation methods (e.g., FiLM) and direct triplane linear interpolation, and demonstrate clear advantages. In summary, LIM is the first feed-forward model capable of high-speed tracked 4D asset reconstruction across diverse categories.

This is the 1st part of the dissertation for my master degree and compares the power consumption using the default floating point (32bit) and Nvidia mixed precision (16bit and 32bit) while training a classification ML model. A custom PC with specific hardware was built to perform the experiments, and different ML hyper-parameters, such as batch size, neurons, and epochs, were chosen to build Deep Neural Networks (DNN). Additionally, various software was used during the experiments to collect the power consumption data in Watts from the Graphics Processing Unit (GPU), Central Processing Unit (CPU), Random Access Memory (RAM) and manually from a wattmeter connected to the wall. A benchmarking test with default hyper parameter values for the DNN was used as a reference, while the experiments used a combination of different settings. The results were recorded in Excel, and descriptive statistics were chosen to calculate the mean between the groups and compare them using graphs and tables. The outcome was positive when using mixed precision combined with specific hyper-parameters. Compared to the benchmarking, the optimisation for the classification reduced the power consumption between 7 and 11 Watts. Similarly, the carbon footprint is reduced because the calculation uses the same power consumption data. Still, a consideration is required when configuring hyper-parameters because it can negatively affect hardware performance. However, this research required inferential statistics, specifically ANOVA and T-test, to compare the relationship between the means. Furthermore, tests indicated no statistical significance of the relationship between the benchmarking and experiments. However, a more extensive implementation with a cluster of GPUs can increase the sample size significantly, as it is an essential factor and can change the outcome of the statistical analysis.

We study nonparametric density estimation in non-stationary drift settings. Given a sequence of independent samples taken from a distribution that gradually changes in time, the goal is to compute the best estimate for the current distribution. We prove tight minimax risk bounds for both discrete and continuous smooth densities, where the minimum is over all possible estimates and the maximum is over all possible distributions that satisfy the drift constraints. Our technique handles a broad class of drift models, and generalizes previous results on agnostic learning under drift.

We present a comprehensive treatment of relative oscillation theory for finite Jacobi matrices. We show that the difference of the number of eigenvalues of two Jacobi matrices in an interval equals the number of weighted sign-changes of the Wronskian of suitable solutions of the two underlying difference equations. Until now only the case of perturbations of the main diagonal was known. We extend the known results to arbitrary perturbations, allow any (half-)open and closed spectral intervals, simplify the proof, and establish the comparison theorem.

Solving Bayesian inverse problems efficiently remains a significant challenge due to the complexity of posterior distributions and the computational cost of traditional sampling methods. Given a series of observations and the forward model, we want to recover the distribution of the parameters, conditioned on observed experimental data. We show, that combining Conditional Flow Mathching (CFM) with transformer-based architecture, we can efficiently sample from such kind of distribution, conditioned on variable number of observations.

We propose Gaussian Frosting, a novel mesh-based representation for high-quality rendering and editing of complex 3D effects in real-time. Our approach builds on the recent 3D Gaussian Splatting framework, which optimizes a set of 3D Gaussians to approximate a radiance field from images. We propose first extracting a base mesh from Gaussians during optimization, then building and refining an adaptive layer of Gaussians with a variable thickness around the mesh to better capture the fine details and volumetric effects near the surface, such as hair or grass. We call this layer Gaussian Frosting, as it resembles a coating of frosting on a cake. The fuzzier the material, the thicker the frosting. We also introduce a parameterization of the Gaussians to enforce them to stay inside the frosting layer and automatically adjust their parameters when deforming, rescaling, editing or animating the mesh. Our representation allows for efficient rendering using Gaussian splatting, as well as editing and animation by modifying the base mesh. We demonstrate the effectiveness of our method on various synthetic and real scenes, and show that it outperforms existing surface-based approaches. We will release our code and a web-based viewer as additional contributions. Our project page is the following: https://anttwo.github.io/frosting/

We present a neural framework for learning conditional optimal transport (OT) maps between probability distributions. Our approach introduces a conditioning mechanism capable of processing both categorical and continuous conditioning variables simultaneously. At the core of our method lies a hypernetwork that generates transport layer parameters based on these inputs, creating adaptive mappings that outperform simpler conditioning methods. Comprehensive ablation studies demonstrate the superior performance of our method over baseline configurations. Furthermore, we showcase an application to global sensitivity analysis, offering high performance in computing OT-based sensitivity indices. This work advances the state-of-the-art in conditional optimal transport, enabling broader application of optimal transport principles to complex, high-dimensional domains such as generative modeling and black-box model explainability.

Conditional normalizing flows can generate diverse image samples for solving inverse problems. Most normalizing flows for inverse problems in imaging employ the conditional affine coupling layer that can generate diverse images quickly. However, unintended severe artifacts are occasionally observed in the output of them. In this work, we address this critical issue by investigating the origins of these artifacts and proposing the conditions to avoid them. First of all, we empirically and theoretically reveal that these problems are caused by "exploding inverse" in the conditional affine coupling layer for certain out-of-distribution (OOD) conditional inputs. Then, we further validated that the probability of causing erroneous artifacts in pixels is highly correlated with a Mahalanobis distance-based OOD score for inverse problems in imaging. Lastly, based on our investigations, we propose a remark to avoid exploding inverse and then based on it, we suggest a simple remedy that substitutes the affine coupling layers with the modified rational quadratic spline coupling layers in normalizing flows, to encourage the robustness of generated image samples. Our experimental results demonstrated that our suggested methods effectively suppressed critical artifacts occurring in normalizing flows for super-resolution space generation and low-light image enhancement.

In this paper, we revisit the problem of sparse linear regression in the local differential privacy (LDP) model. Existing research in the non-interactive and sequentially local models has focused on obtaining the lower bounds for the case where the underlying parameter is 1-sparse, and extending such bounds to the more general k-sparse case has proven to be challenging. Moreover, it is unclear whether efficient non-interactive LDP (NLDP) algorithms exist. To address these issues, we first consider the problem in the epsilon non-interactive LDP model and provide a lower bound of Omega(sqrt{dklog d}{nepsilon}) on the ell_2-norm estimation error for sub-Gaussian data, where n is the sample size and d is the dimension of the space. We propose an innovative NLDP algorithm, the very first of its kind for the problem. As a remarkable outcome, this algorithm also yields a novel and highly efficient estimator as a valuable by-product. Our algorithm achieves an upper bound of O({dsqrt{k}{nepsilon}}) for the estimation error when the data is sub-Gaussian, which can be further improved by a factor of O(d) if the server has additional public but unlabeled data. For the sequentially interactive LDP model, we show a similar lower bound of Omega({sqrt{dk}{nepsilon}}). As for the upper bound, we rectify a previous method and show that it is possible to achieve a bound of O(ksqrt{d}{nepsilon}). Our findings reveal fundamental differences between the non-private case, central DP model, and local DP model in the sparse linear regression problem.

Quantum algorithms for optimization problems are of general interest. Despite recent progress in classical lower bounds for nonconvex optimization under different settings and quantum lower bounds for convex optimization, quantum lower bounds for nonconvex optimization are still widely open. In this paper, we conduct a systematic study of quantum query lower bounds on finding epsilon-approximate stationary points of nonconvex functions, and we consider the following two important settings: 1) having access to p-th order derivatives; or 2) having access to stochastic gradients. The classical query lower bounds is Omegabig(epsilon^{-1+p{p}}big) regarding the first setting, and Omega(epsilon^{-4}) regarding the second setting (or Omega(epsilon^{-3}) if the stochastic gradient function is mean-squared smooth). In this paper, we extend all these classical lower bounds to the quantum setting. They match the classical algorithmic results respectively, demonstrating that there is no quantum speedup for finding epsilon-stationary points of nonconvex functions with p-th order derivative inputs or stochastic gradient inputs, whether with or without the mean-squared smoothness assumption. Technically, our quantum lower bounds are obtained by showing that the sequential nature of classical hard instances in all these settings also applies to quantum queries, preventing any quantum speedup other than revealing information of the stationary points sequentially.

Conformal prediction is a theoretically grounded framework for constructing predictive intervals. We study conformal prediction with missing values in the covariates -- a setting that brings new challenges to uncertainty quantification. We first show that the marginal coverage guarantee of conformal prediction holds on imputed data for any missingness distribution and almost all imputation functions. However, we emphasize that the average coverage varies depending on the pattern of missing values: conformal methods tend to construct prediction intervals that under-cover the response conditionally to some missing patterns. This motivates our novel generalized conformalized quantile regression framework, missing data augmentation, which yields prediction intervals that are valid conditionally to the patterns of missing values, despite their exponential number. We then show that a universally consistent quantile regression algorithm trained on the imputed data is Bayes optimal for the pinball risk, thus achieving valid coverage conditionally to any given data point. Moreover, we examine the case of a linear model, which demonstrates the importance of our proposal in overcoming the heteroskedasticity induced by missing values. Using synthetic and data from critical care, we corroborate our theory and report improved performance of our methods.

Neural Radiance Fields (NeRFs) have demonstrated the remarkable potential of neural networks to capture the intricacies of 3D objects. By encoding the shape and color information within neural network weights, NeRFs excel at producing strikingly sharp novel views of 3D objects. Recently, numerous generalizations of NeRFs utilizing generative models have emerged, expanding its versatility. In contrast, Gaussian Splatting (GS) offers a similar render quality with faster training and inference as it does not need neural networks to work. It encodes information about the 3D objects in the set of Gaussian distributions that can be rendered in 3D similarly to classical meshes. Unfortunately, GS are difficult to condition since they usually require circa hundred thousand Gaussian components. To mitigate the caveats of both models, we propose a hybrid model Viewing Direction Gaussian Splatting (VDGS) that uses GS representation of the 3D object's shape and NeRF-based encoding of color and opacity. Our model uses Gaussian distributions with trainable positions (i.e. means of Gaussian), shape (i.e. covariance of Gaussian), color and opacity, and a neural network that takes Gaussian parameters and viewing direction to produce changes in the said color and opacity. As a result, our model better describes shadows, light reflections, and the transparency of 3D objects without adding additional texture and light components.

Even when massively overparameterized, deep neural networks show a remarkable ability to generalize. Research on this phenomenon has focused on generalization within distribution, via smooth interpolation. Yet in some settings neural networks also learn to extrapolate to data far beyond the bounds of the original training set, sometimes even allowing for infinite generalization, implying that an algorithm capable of solving the task has been learned. Here we undertake a case study of the learning dynamics of recurrent neural networks (RNNs) trained on the streaming parity task in order to develop an effective theory of algorithm development. The streaming parity task is a simple but nonlinear task defined on sequences up to arbitrary length. We show that, with sufficient finite training experience, RNNs exhibit a phase transition to perfect infinite generalization. Using an effective theory for the representational dynamics, we find an implicit representational merger effect which can be interpreted as the construction of a finite automaton that reproduces the task. Overall, our results disclose one mechanism by which neural networks can generalize infinitely from finite training experience.

Control variates are variance reduction tools for Monte Carlo estimators. They can provide significant variance reduction, but usually require a large number of samples, which can be prohibitive when sampling or evaluating the integrand is computationally expensive. Furthermore, there are many scenarios where we need to compute multiple related integrals simultaneously or sequentially, which can further exacerbate computational costs. In this paper, we propose vector-valued control variates, an extension of control variates which can be used to reduce the variance of multiple Monte Carlo estimators jointly. This allows for the transfer of information across integration tasks, and hence reduces the need for a large number of samples. We focus on control variates based on kernel interpolants and our novel construction is obtained through a generalised Stein identity and the development of novel matrix-valued Stein reproducing kernels. We demonstrate our methodology on a range of problems including multifidelity modelling, Bayesian inference for dynamical systems, and model evidence computation through thermodynamic integration.

In this note we establish a 1-to-1 correspondence between the class of generalized quadrangles with ovoids and the class of balanced incomplete block designs that posses a non-triangular local resolution system and have the appropriate parameters. We present a non-triangular local resolution system for a difference family BIBD construction of Sprott.

We study the problem of efficiently generating differentially private synthetic data that approximate the statistical properties of an underlying sensitive dataset. In recent years, there has been a growing line of work that approaches this problem using first-order optimization techniques. However, such techniques are restricted to optimizing differentiable objectives only, severely limiting the types of analyses that can be conducted. For example, first-order mechanisms have been primarily successful in approximating statistical queries only in the form of marginals for discrete data domains. In some cases, one can circumvent such issues by relaxing the task's objective to maintain differentiability. However, even when possible, these approaches impose a fundamental limitation in which modifications to the minimization problem become additional sources of error. Therefore, we propose Private-GSD, a private genetic algorithm based on zeroth-order optimization heuristics that do not require modifying the original objective. As a result, it avoids the aforementioned limitations of first-order optimization. We empirically evaluate Private-GSD against baseline algorithms on data derived from the American Community Survey across a variety of statistics--otherwise known as statistical queries--both for discrete and real-valued attributes. We show that Private-GSD outperforms the state-of-the-art methods on non-differential queries while matching accuracy in approximating differentiable ones.

The notions of disintegration and Bayesian inversion are fundamental in conditional probability theory. They produce channels, as conditional probabilities, from a joint state, or from an already given channel (in opposite direction). These notions exist in the literature, in concrete situations, but are presented here in abstract graphical formulations. The resulting abstract descriptions are used for proving basic results in conditional probability theory. The existence of disintegration and Bayesian inversion is discussed for discrete probability, and also for measure-theoretic probability --- via standard Borel spaces and via likelihoods. Finally, the usefulness of disintegration and Bayesian inversion is illustrated in several examples.

Trajectory planning is commonly used as part of a local planner in autonomous driving. This paper considers the problem of planning a continuous-curvature-rate trajectory between fixed start and goal states that minimizes a tunable trade-off between passenger comfort and travel time. The problem is an instance of infinite dimensional optimization over two continuous functions: a path, and a velocity profile. We propose a simplification of this problem that facilitates the discretization of both functions. This paper also proposes a method to quickly generate minimal-length paths between start and goal states based on a single tuning parameter: the second derivative of curvature. Furthermore, we discretize the set of velocity profiles along a given path into a selection of acceleration way-points along the path. Gradient-descent is then employed to minimize cost over feasible choices of the second derivative of curvature, and acceleration way-points, resulting in a method that repeatedly solves the path and velocity profiles in an iterative fashion. Numerical examples are provided to illustrate the benefits of the proposed methods.

Given a factorization of an image into a sum of linear components, we present a zero-shot method to control each individual component through diffusion model sampling. For example, we can decompose an image into low and high spatial frequencies and condition these components on different text prompts. This produces hybrid images, which change appearance depending on viewing distance. By decomposing an image into three frequency subbands, we can generate hybrid images with three prompts. We also use a decomposition into grayscale and color components to produce images whose appearance changes when they are viewed in grayscale, a phenomena that naturally occurs under dim lighting. And we explore a decomposition by a motion blur kernel, which produces images that change appearance under motion blurring. Our method works by denoising with a composite noise estimate, built from the components of noise estimates conditioned on different prompts. We also show that for certain decompositions, our method recovers prior approaches to compositional generation and spatial control. Finally, we show that we can extend our approach to generate hybrid images from real images. We do this by holding one component fixed and generating the remaining components, effectively solving an inverse problem.

We present IntPhys 2, a video benchmark designed to evaluate the intuitive physics understanding of deep learning models. Building on the original IntPhys benchmark, IntPhys 2 focuses on four core principles related to macroscopic objects: Permanence, Immutability, Spatio-Temporal Continuity, and Solidity. These conditions are inspired by research into intuitive physical understanding emerging during early childhood. IntPhys 2 offers a comprehensive suite of tests, based on the violation of expectation framework, that challenge models to differentiate between possible and impossible events within controlled and diverse virtual environments. Alongside the benchmark, we provide performance evaluations of several state-of-the-art models. Our findings indicate that while these models demonstrate basic visual understanding, they face significant challenges in grasping intuitive physics across the four principles in complex scenes, with most models performing at chance levels (50%), in stark contrast to human performance, which achieves near-perfect accuracy. This underscores the gap between current models and human-like intuitive physics understanding, highlighting the need for advancements in model architectures and training methodologies.

We present a faster algorithm to generate a warm start for sampling an arbitrary logconcave density specified by an evaluation oracle, leading to the first sub-cubic sampling algorithms for inputs in (near-)isotropic position. A long line of prior work incurred a warm-start penalty of at least linear in the dimension, hitting a cubic barrier, even for the special case of uniform sampling from convex bodies. Our improvement relies on two key ingredients of independent interest. (1) We show how to sample given a warm start in weaker notions of distance, in particular q-R\'enyi divergence for q=mathcal{O}(1), whereas previous analyses required stringent infty-R\'enyi divergence (with the exception of Hit-and-Run, whose known mixing time is higher). This marks the first improvement in the required warmness since Lov\'asz and Simonovits (1991). (2) We refine and generalize the log-Sobolev inequality of Lee and Vempala (2018), originally established for isotropic logconcave distributions in terms of the diameter of the support, to logconcave distributions in terms of a geometric average of the support diameter and the largest eigenvalue of the covariance matrix.

Solving a linear system Ax=b is a fundamental scientific computing primitive for which numerous solvers and preconditioners have been developed. These come with parameters whose optimal values depend on the system being solved and are often impossible or too expensive to identify; thus in practice sub-optimal heuristics are used. We consider the common setting in which many related linear systems need to be solved, e.g. during a single numerical simulation. In this scenario, can we sequentially choose parameters that attain a near-optimal overall number of iterations, without extra matrix computations? We answer in the affirmative for Successive Over-Relaxation (SOR), a standard solver whose parameter omega has a strong impact on its runtime. For this method, we prove that a bandit online learning algorithm--using only the number of iterations as feedback--can select parameters for a sequence of instances such that the overall cost approaches that of the best fixed omega as the sequence length increases. Furthermore, when given additional structural information, we show that a contextual bandit method asymptotically achieves the performance of the instance-optimal policy, which selects the best omega for each instance. Our work provides the first learning-theoretic treatment of high-precision linear system solvers and the first end-to-end guarantees for data-driven scientific computing, demonstrating theoretically the potential to speed up numerical methods using well-understood learning algorithms.

We address the problem of detecting and erasing furniture from a wide angle photograph of a room. Inpainting large regions of an indoor scene often results in geometric inconsistencies of background elements within the inpaint mask. To address this problem, we utilize perceptual information (e.g. instance segmentation, and room layout) to produce a geometrically consistent empty version of a room. We share important details to make this system viable, such as per-plane inpainting, automatic rectification, and texture refinement. We provide detailed ablation along with qualitative examples, justifying our design choices. We show an application of our system by removing real furniture from a room and redecorating it with virtual furniture.

We extend conformal prediction to control the expected value of any monotone loss function. The algorithm generalizes split conformal prediction together with its coverage guarantee. Like conformal prediction, the conformal risk control procedure is tight up to an O(1/n) factor. We also introduce extensions of the idea to distribution shift, quantile risk control, multiple and adversarial risk control, and expectations of U-statistics. Worked examples from computer vision and natural language processing demonstrate the usage of our algorithm to bound the false negative rate, graph distance, and token-level F1-score.

Gradient clipping is a popular modification to standard (stochastic) gradient descent, at every iteration limiting the gradient norm to a certain value c >0. It is widely used for example for stabilizing the training of deep learning models (Goodfellow et al., 2016), or for enforcing differential privacy (Abadi et al., 2016). Despite popularity and simplicity of the clipping mechanism, its convergence guarantees often require specific values of c and strong noise assumptions. In this paper, we give convergence guarantees that show precise dependence on arbitrary clipping thresholds c and show that our guarantees are tight with both deterministic and stochastic gradients. In particular, we show that (i) for deterministic gradient descent, the clipping threshold only affects the higher-order terms of convergence, (ii) in the stochastic setting convergence to the true optimum cannot be guaranteed under the standard noise assumption, even under arbitrary small step-sizes. We give matching upper and lower bounds for convergence of the gradient norm when running clipped SGD, and illustrate these results with experiments.

We study the connection between multicalibration and boosting for squared error regression. First we prove a useful characterization of multicalibration in terms of a ``swap regret'' like condition on squared error. Using this characterization, we give an exceedingly simple algorithm that can be analyzed both as a boosting algorithm for regression and as a multicalibration algorithm for a class H that makes use only of a standard squared error regression oracle for H. We give a weak learning assumption on H that ensures convergence to Bayes optimality without the need to make any realizability assumptions -- giving us an agnostic boosting algorithm for regression. We then show that our weak learning assumption on H is both necessary and sufficient for multicalibration with respect to H to imply Bayes optimality. We also show that if H satisfies our weak learning condition relative to another class C then multicalibration with respect to H implies multicalibration with respect to C. Finally we investigate the empirical performance of our algorithm experimentally using an open source implementation that we make available. Our code repository can be found at https://github.com/Declancharrison/Level-Set-Boosting.

We use tools from geometric statistics to analyze the usual estimation procedure of a template shape. This applies to shapes from landmarks, curves, surfaces, images etc. We demonstrate the asymptotic bias of the template shape estimation using the stratified geometry of the shape space. We give a Taylor expansion of the bias with respect to a parameter sigma describing the measurement error on the data. We propose two bootstrap procedures that quantify the bias and correct it, if needed. They are applicable for any type of shape data. We give a rule of thumb to provide intuition on whether the bias has to be corrected. This exhibits the parameters that control the bias' magnitude. We illustrate our results on simulated and real shape data.

We study the behavior of deterministic methods for solving inverse problems in imaging. These methods are commonly designed to achieve two goals: (1) attaining high perceptual quality, and (2) generating reconstructions that are consistent with the measurements. We provide a rigorous proof that the better a predictor satisfies these two requirements, the larger its Lipschitz constant must be, regardless of the nature of the degradation involved. In particular, to approach perfect perceptual quality and perfect consistency, the Lipschitz constant of the model must grow to infinity. This implies that such methods are necessarily more susceptible to adversarial attacks. We demonstrate our theory on single image super-resolution algorithms, addressing both noisy and noiseless settings. We also show how this undesired behavior can be leveraged to explore the posterior distribution, thereby allowing the deterministic model to imitate stochastic methods.

Overparametrization often helps improve the generalization performance. This paper proposes a dual view of overparametrization suggesting that downsampling may also help generalize. Motivated by this dual view, we characterize two out-of-sample prediction risks of the sketched ridgeless least square estimator in the proportional regime masymp n asymp p, where m is the sketching size, n the sample size, and p the feature dimensionality. Our results reveal the statistical role of downsampling. Specifically, downsampling does not always hurt the generalization performance, and may actually help improve it in some cases. We identify the optimal sketching sizes that minimize the out-of-sample prediction risks, and find that the optimally sketched estimator has stabler risk curves that eliminates the peaks of those for the full-sample estimator. We then propose a practical procedure to empirically identify the optimal sketching size. Finally, we extend our results to cover central limit theorems and misspecified models. Numerical studies strongly support our theory.

Split conformal prediction has recently sparked great interest due to its ability to provide formally guaranteed uncertainty sets or intervals for predictions made by black-box neural models, ensuring a predefined probability of containing the actual ground truth. While the original formulation assumes data exchangeability, some extensions handle non-exchangeable data, which is often the case in many real-world scenarios. In parallel, some progress has been made in conformal methods that provide statistical guarantees for a broader range of objectives, such as bounding the best F_1-score or minimizing the false negative rate in expectation. In this paper, we leverage and extend these two lines of work by proposing non-exchangeable conformal risk control, which allows controlling the expected value of any monotone loss function when the data is not exchangeable. Our framework is flexible, makes very few assumptions, and allows weighting the data based on its relevance for a given test example; a careful choice of weights may result on tighter bounds, making our framework useful in the presence of change points, time series, or other forms of distribution drift. Experiments with both synthetic and real world data show the usefulness of our method.

Consider a random uniform sample of n points in a compact region A of Euclidean d-space, d geq 2, with a smooth or (when d=2) polygonal boundary. Fix k bf N. Let T_{n,k} be the threshold r at which the geometric graph on these n vertices with distance parameter r becomes k-connected. We show that if d=2 then n (pi/|A|) T_{n,1}^2 - log n is asymptotically standard Gumbel. For (d,k) neq (2,1), it is n (theta_d/|A|) T_{n,k}^d - (2-2/d) log n - (4-2k-2/d) log log n that converges in distribution to a nondegenerate limit, where theta_d is the volume of the unit ball. The limit is Gumbel with scale parameter 2 except when (d,k)=(2,2) where the limit is two component extreme value distributed. The different cases reflect the fact that boundary effects are more more important in some cases than others. We also give similar results for the largest k-nearest neighbour link U_{n,k} in the sample, and show T_{n,k}=U_{n,k} with high probability. We provide estimates on rates of convergence and give similar results for Poisson samples in A. Finally, we give similar results even for non-uniform samples, with a less explicit sequence of centring constants.