Bounds on the Spectral Sparsification of Symmetric and Off-Diagonal Nonnegative Real Matrices

TL;DR

This paper demonstrates that any off-diagonal nonnegative symmetric matrix can be approximated spectrally by a sparse, nonnegative symmetric matrix, advancing spectral sparsification techniques for such matrices.

Contribution

It introduces a method to construct sparse, spectrally close approximations for off-diagonal nonnegative symmetric matrices, a novel extension in spectral sparsification.

Findings

Existence of sparse, spectrally close approximations for off-diagonal nonnegative symmetric matrices

Spectral closeness achieved with nonnegative symmetric matrices

Potential applications in graph sparsification and matrix approximation

Abstract

We say that a square real matrix $M$ is \emph{off-diagonal nonnegative} if and only if all entries outside its diagonal are nonnegative real numbers. In this note we show that for any off-diagonal nonnegative symmetric matrix $M$, there exists a nonnegative symmetric matrix $\widehat{M}$ which is sparse and close in spectrum to $M$.