Stability and complexity of mixed discriminants

TL;DR

This paper presents a quasi-polynomial time algorithm for approximating the mixed discriminant of certain matrices, extending to sums of principal minors, with implications for computational complexity in matrix analysis.

Contribution

It introduces a quasi-polynomial approximation algorithm for mixed discriminants of positive semidefinite matrices under specific conditions, and extends to sums of principal minors, linking to complexity results.

Findings

Approximation within relative error in quasi-polynomial time for matrices close to identity.

Extension of results to doubly stochastic matrix tuples using root bounds.

Quasi-polynomial algorithm for sums of principal minors when operator norm is less than 1.

Abstract

We show that the mixed discriminant of $n$ positive semidefinite $n \times n$ real symmetric matrices can be approximated within a relative error $\epsilon >0$ in quasi-polynomial $n^{O(\ln n -\ln \epsilon)}$ time, provided the distance of each matrix to the identity matrix in the operator norm does not exceed some absolute constant $\gamma_0 >0$. We deduce a similar result for the mixed discriminant of doubly stochastic $n$-tuples of matrices from the Marcus - Spielman - Srivastava bound on the roots of the mixed characteristic polynomial. Finally, we construct a quasi-polynomial algorithm for approximating the sum of $m$-th powers of principal minors of a matrix, provided the operator norm of the matrix is strictly less than 1. As is shown by Gurvits, for $m=2$ the problem is $\#P$-hard and covers the problem of computing the mixed discriminant of positive semidefinite matrices of rank 2.