Bipartite Graph Approximation by Eigenvalue Optimization

Abstract

Graphs are a powerful tool for representing entities and their relationships. Current advances in graph signal processing have made it possible to analyze graph-based data more effectively. Recent research show that, to ensure critical sampling, manyfilterbank design algorithms are only applicable to bipartite graphs. However, general graph signals may not exist on a bipartite graph structure. To overcome this difficulty, we propose in this paper a novel algorithm to find a bipartite approximation to the original non-bipartite graph while preserving its global structure. To achieve this goal, the original bipartite graph approximation (BGA) problem is constructed based on eigenvalue optimization of adjacency matrix, which is then relaxed so as to obtain a closed-form solution. We introduce the alternating direction method of multipliers (ADMM) to achieve a single bipartite graph or a set of edge-disjoint bipartite subgraphs that approximates the original graph. Additionally, we develop a distributed version of the BGA to address the computational challenges when processing large-scale graphs. Experimental results demonstrate the effectiveness of the proposed method and suggest it as a promising alternative approach for bipartite graph decomposition.