A Matrix Hyperbolic Cosine Algorithm and Applications

TL;DR

This paper extends Spencer's hyperbolic cosine algorithm to matrices, enabling efficient deterministic algorithms for constructing Cayley graphs and spectral sparsification, with applications to graph theory and matrix approximation.

Contribution

It introduces a matrix hyperbolic cosine algorithm and applies it to develop deterministic algorithms for Cayley graph construction and spectral sparsification.

Findings

Deterministic construction of expanding Cayley graphs in near-optimal time.

Fast deterministic spectral sparsification algorithms for positive semi-definite matrices.

Improved element-wise sparsification methods for diagonally dominant-like matrices.

Abstract

In this paper, we generalize Spencer's hyperbolic cosine algorithm to the matrix-valued setting. We apply the proposed algorithm to several problems by analyzing its computational efficiency under two special cases of matrices; one in which the matrices have a group structure and an other in which they have rank-one. As an application of the former case, we present a deterministic algorithm that, given the multiplication table of a finite group of size $n$, it constructs an expanding Cayley graph of logarithmic degree in near-optimal O(n^2 log^3 n) time. For the latter case, we present a fast deterministic algorithm for spectral sparsification of positive semi-definite matrices, which implies an improved deterministic algorithm for spectral graph sparsification of dense graphs. In addition, we give an elementary connection between spectral sparsification of positive semi-definite matrices and element-wise matrix sparsification. As a consequence, we obtain improved element-wise sparsification algorithms for diagonally dominant-like matrices.