is composed of two modules: the graph learning module and node feature learning module. The graph learning module learns a graph encoded by an $N \times N$ dense adjacency matrix $\mathbf{A}$ from input node features. $\mathbf{A}$ is then employed by the node feature learning module to propagate node features via our proposed graph convolution in Eq. . Both modules are jointly optimized according to the objective in Eq. , *i.e.*, the loss function is defined as $$\begin{split}
E=& -\sum_{i=1}^N \sum_{c=1}^C \mathbf{Y}_{ic} \log(\hat{\mathbf{Y}}_{ic}(\{\mathbf{W}_l, \mathbf{R}_l\})) \\
& + \lambda \sum_l \sum_{\{i,j\}}\exp\left\{-\|\mathbf{R}^{\top}_l (\mathbf{f}^i_l - \mathbf{f}^j_l)\|_2^2 \right\} \, s_{i,j},
\end{split}
\label{eq:loss}$$ The algorithmic details are provided in Alg. \[alg:layer\].

**Input:** features $\mathbf{F}_{l-1}$ and adjacency matrix $\mathbf{A}_{l-1}$ from the previous layer\
**Output:** features $\mathbf{F}_l$ and learned adjacency matrix $\mathbf{A}_l$\
**Initialize:** trainable parameters $\mathbf{R}_l, \mathbf{W}_l$

**Forward Pass:** Compute distance metric $d_\mathbf{M}$ by Eq. Update $\mathbf{A}_l $ by Eq. Update $\mathbf{F}_l $ by Eq. **Backward Pass:** Update $\mathbf{R}_l$ and $\mathbf{W}_l$ via backward propagation with the loss function defined in Eq. .

![image](figures/network.pdf){width="6in"}

JLGCN Network
-------------

In principle, our JLGCN layer can be integrated into any GCNN architecture for both node classification and graph classification tasks. Fig. \[fig:network\] demonstrates the architecture of the JLGCN network. Node features are 