Some algorithms for maximum volume and cross approximation of symmetric semidefinite matrices

TL;DR

This paper introduces a new greedy algorithm for maximum volume submatrix selection in symmetric positive semidefinite matrices, providing bounds on approximation error and efficient deterministic algorithms for cross approximation.

Contribution

The paper proposes a cost-efficient greedy algorithm for maximum volume submatrices in SPSD matrices and establishes error bounds for cross approximation based on principal submatrices.

Findings

The greedy algorithm operates in O(n) time for SPSD matrices.

Any SPSD matrix admits a cross approximation with bounded error relative to the best rank r approximation.

Deterministic algorithms can achieve quasi-optimal cross approximation with O(n^3) complexity.

Abstract

Finding the $r\times r$ submatrix of maximum volume of a matrix $A\in\mathbb R^{n\times n}$ is an NP hard problem that arises in a variety of applications. We propose a new greedy algorithm of cost $\mathcal O(n)$, for the case $A$ symmetric positive semidefinite (SPSD) and we discuss its extension to related optimization problems such as the maximum ratio of volumes. In the second part of the paper we prove that any SPSD matrix admits a cross approximation built on a principal submatrix whose approximation error is bounded by $(r+1)$ times the error of the best rank $r$ approximation in the nuclear norm. In the spirit of recent work by Cortinovis and Kressner we derive some deterministic algorithms which are capable to retrieve a quasi optimal cross approximation with cost $\mathcal O(n^3)$.