Low rank approximation of positive semi-definite symmetric matrices using Gaussian elimination and volume sampling

TL;DR

This paper explores low-rank approximation techniques for positive semi-definite matrices using Gaussian elimination and volume sampling, providing explicit error formulas and bounds to improve computational efficiency.

Contribution

It introduces invariant matrix theory to derive explicit error formulas for low-rank approximations based on volume sampling, enhancing understanding of approximation errors.

Findings

Explicit error formulas for volume sampling approximations

Bounds on approximation error based on eigenvalues

Examples with exact expected error norms

Abstract

Positive semi-definite matrices commonly occur as normal matrices of least squares problems in statistics or as kernel matrices in machine learning and approximation theory. They are typically large and dense. Thus algorithms to solve systems with such a matrix can be very costly. A core idea to reduce computational complexity is to approximate the matrix by one with a low rank. The optimal and well understood choice is based on the eigenvalue decomposition of the matrix. Unfortunately, this is computationally very expensive. Cheaper methods are based on Gaussian elimination but they require pivoting. We will show how invariant matrix theory provides explicit error formulas for an averaged error based on volume sampling. The formula leads to ratios of elementary symmetric polynomials on the eigenvalues. We discuss some new an old bounds and include several examples where an expected error norm can be computed exactly.