 learns from objects without part annotations. $$\begin{split}
S^{\textrm{unant}}_{\textrm{AOG}}&={\sum}_{u\in Child(v^{*})}S^{\textrm{unant}}_{u}\\
L^{\textrm{unant}}({\boldsymbol\Lambda}_{\textrm{AOG}})&={\sum}_{u\in Child(v^{*})}\lambda^{\textrm{close}}\Vert\Delta{\bf p}_{u}\Vert^2
\end{split}
\label{sec:unsuper}$$ where the first term $S^{\textrm{unant}}_{\textrm{AOG}}$ denotes the inference score at the level of latent patterns without ground-truth annotations of object parts. Please see the appendix for the computation of $S^{\textrm{unant}}_{u}$. The second term $L^{\textrm{unant}}({\boldsymbol\Lambda}_{\textrm{AOG}})$ penalizes latent patterns that are far from their parent $v^{*}$. This loss encourages the assigned neural unit to be close to its parent latent pattern. We assume that 1) latent patterns that frequently appear among unannotated objects may potentially represent stable part appearance and should have higher priorities; and that 2) latent patterns spatially closer to their parent part templates are usually more reliable.

When we set $\lambda_{v^{*}}$ to a constant $\lambda^{\textrm{inf}}\sum_{k=1}^{K}n_{k}$, we can transform the learning objective in Equation (\[eqn:LossAOG\]) as follows. $$\forall v\in Child(top), \quad\max_{{\boldsymbol\theta}_{v}}{\bf L}_{v},\quad {\bf L}_{v}\!=\!\!\!\!\!\!\!\!\sum_{u\in Child(v)}\!\!\!\!\!\!\!Score(u)
\label{eqn:subAOG}$$ where [$Score(u)\!=\!\mathbb{E}_{I\in{\bf I}_{v}}[S_{u}+S^{\textrm{inf}}(\Lambda_{u}|\Lambda^{*}_{v})]$ $+\mathbb{E}_{I'\in{\bf I}^{\textrm{obj}}}$ $\lambda^{\textrm{unant}}[S^{\textrm{unant}}_{u}-\lambda^{\textrm{close}}\Vert\Delta{\bf p}_{u}\Vert^2]$]{}. ${\boldsymbol\theta}_{v}\subset{\boldsymbol\theta}$ denotes the parameters for the sub-AOG of the part template $v$. We use ${\bf I}_{v}\subset{\bf I}^{\textrm{ant}}$ to denote the subset of images that are annotated with $v$ as the ground-truth part template.

**Learning the sub-AOG for each part template:**[` `]{} Based on Equation (\[eqn:subAOG\]), we can mine the sub-AOG for eac