  Graph coarsening is a widely used dimensionality reduction technique for
approaching large-scale graph machine learning problems. Given a large graph,
graph coarsening aims to learn a smaller-tractable graph while preserving the
properties of the originally given graph. Graph data consist of node features
and graph matrix (e.g., adjacency and Laplacian). The existing graph coarsening
methods ignore the node features and rely solely on a graph matrix to simplify
graphs. In this paper, we introduce a novel optimization-based framework for
graph dimensionality reduction. The proposed framework lies in the unification
of graph learning and dimensionality reduction. It takes both the graph matrix
and the node features as the input and learns the coarsen graph matrix and the
coarsen feature matrix jointly while ensuring desired properties. The proposed
optimization formulation is a multi-block non-convex optimization problem,
which is solved efficiently by leveraging block majorization-minimization,
$\log$ determinant, Dirichlet energy, and regularization frameworks. The
proposed algorithms are provably convergent and practically amenable to
numerous tasks. It is also established that the learned coarsened graph is
$\epsilon\in(0,1)$ similar to the original graph. Extensive experiments
elucidate the efficacy of the proposed framework for real-world applications.
