A Unified Framework for Optimization-Based Graph Coarsening

TL;DR

This paper introduces a unified optimization framework for graph coarsening that jointly reduces graph size and preserves properties by considering both node features and graph structure, improving large-scale graph learning.

Contribution

It presents a novel joint optimization approach for graph coarsening that incorporates node features, unifying graph learning and dimensionality reduction in a single framework.

Findings

The framework effectively preserves graph properties in coarsening.

Algorithms are provably convergent and efficient.

Experimental results demonstrate improved performance on real-world tasks.

Abstract

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.