A Matrix Hyperbolic Cosine Algorithm and Applications
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 , 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.
Cite
@article{arxiv.1103.2793,
title = {A Matrix Hyperbolic Cosine Algorithm and Applications},
author = {Anastasios Zouzias},
journal= {arXiv preprint arXiv:1103.2793},
year = {2015}
}
Comments
16 pages, simplified proof and corrected acknowledging of prior work in (current) Section 4