is a real and symmetric matrix, where $a_{i,j} \geq 0$ is the weight assigned to the edge $(i,j)$ connecting vertices $i$ and $j$. Edge weights often measure the similarity between connected vertices.

The graph Laplacian matrix is defined from the adjacency matrix. Among different variants of Laplacian matrices, the *combinatorial graph Laplacian* is defined as $\mathbf{L}:=\mathbf{D}-\mathbf{A}$, where $\mathbf{D}$ is the *degree matrix*—a diagonal matrix where $d_{i,i}=\sum_{j=1}^N a_{i,j}$.

Graph Laplacian Regularizer
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Graph signal refers to data that resides on the nodes of a graph, such as temperatures on a sensor network. A graph signal $\mathbf{x}\in\mathbb{R}^N$ defined on a graph $\mathcal{G}$ is *smooth* with respect to $\mathcal{G}$ [@spielman2007spectral] if $$\mathbf{x}^{\top} \mathbf{L} \mathbf{x} =\sum_{i=1}^{N} \sum_{j=1}^{N} w_{i,j}(x_i - x_j)^2 < \epsilon,
    \label{eq:prior}$$ where $\epsilon$ is a small positive scalar. To satisfy Eq. , connected node pair $x_i$ and $x_j$ must be similar for a large edge weight $w_{i,j}$; for a small $w_{i,j}$, $x_i$ and $x_j$ can differ significantly. Hence, Eq. forces $\mathbf{x}$ to adapt to the topology of $\mathcal{G}$, and is commonly called the *graph Laplacian Regularizer* (GLR) [@shuman2013emerging; @pang2017graph].

Joint Learning of Graphs and Features {#sec:method}
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In this section, we first propose the problem formulations of graph learning and node feature 