Singular values of Gaussian matrices and permanent estimators

TL;DR

This paper provides bounds on the small singular values of Gaussian matrices with inhomogeneous variances and demonstrates that certain graph-based permanent estimators achieve high-probability sub-exponential accuracy.

Contribution

It introduces new estimates for singular values of Gaussian matrices with variance profiles and proves the effectiveness of a permanent estimator for graphs with expansion properties.

Findings

Singular value estimates for Gaussian matrices with broad-connectedness

High-probability sub-exponential error bounds for the permanent estimator

Applicability to graphs satisfying expansion conditions

Abstract

We present estimates on the small singular values of a class of matrices with independent Gaussian entries and inhomogeneous variance profile, satisfying a broad-connectedness condition. Using these estimates and concentration of measure for the spectrum of Gaussian matrices with independent entries, we prove that for a large class of graphs satisfying an appropriate expansion property, the Barvinok--Godsil-Gutman estimator for the permanent achieves sub-exponential errors with high probability.