Tractable semidefinite bounds of positive maximal singular values

TL;DR

This paper introduces a hierarchy of semidefinite relaxations to efficiently compute certified upper bounds for the positive maximal singular value of matrices, leveraging an extension of Pólya's theorem for better approximation control.

Contribution

It develops a novel hierarchy of semidefinite relaxations based on Pólya's theorem extension, enabling precise approximation of the positive maximal singular value.

Findings

Provides a hierarchy of relaxations that approximate the PMSV as closely as desired.

Uses an extension of Pólya's theorem to decompose positive polynomials into sums of squares.

Controls the size of relaxations by fixing the degree of monomials in the decomposition.

Abstract

We focus on computing certified upper bounds for the positive maximal singular value (PMSV) of a given matrix. The PMSV problem boils down to maximizing a quadratic polynomial on the intersection of the unit sphere and the nonnegative orthant. We provide a hierarchy of tractable semidefinite relaxations to approximate the value of the latter polynomial optimization problem as closely as desired. This hierarchy is based on an extension of P\'olya's representation theorem. Doing so, positive polynomials can be decomposed as weighted sums of squares of $s$-nomials, where $s$ can be a priori fixed ($s=1$ corresponds to monomials, $s=2$ corresponds to binomials, etc.). This in turn allows us to control the size of the resulting semidefinite relaxations.