Sparse Sums of Positive Semidefinite Matrices

TL;DR

This paper develops algorithms for sparsifying sums of positive semidefinite matrices of arbitrary rank, enabling applications in graph sparsification, hypergraph sparsification, and sparse semidefinite programming solutions.

Contribution

It introduces new algorithms based on existing methods for sparsifying positive semidefinite matrix sums, extending previous work beyond rank-one matrices.

Findings

Algorithms effectively sparsify positive semidefinite matrices

Applications include graph and hypergraph sparsifiers

Enables sparse solutions to semidefinite programs

Abstract

Recently there has been much interest in "sparsifying" sums of rank one matrices: modifying the coefficients such that only a few are nonzero, while approximately preserving the matrix that results from the sum. Results of this sort have found applications in many different areas, including sparsifying graphs. In this paper we consider the more general problem of sparsifying sums of positive semidefinite matrices that have arbitrary rank. We give several algorithms for solving this problem. The first algorithm is based on the method of Batson, Spielman and Srivastava (2009). The second algorithm is based on the matrix multiplicative weights update method of Arora and Kale (2007). We also highlight an interesting connection between these two algorithms. Our algorithms have numerous applications. We show how they can be used to construct graph sparsifiers with auxiliary constraints, sparsifiers of hypergraphs, and sparse solutions to semidefinite programs.