A Note on Error Bounds for Pseudo Skeleton Approximations of Matrices

TL;DR

This paper improves error bounds for pseudo skeleton matrix approximations, especially for matrices with polynomially decreasing eigenvalues, enhancing understanding of their accuracy in practical scenarios.

Contribution

It provides new, tighter error bounds for Gaussian elimination-based skeleton approximations, particularly for matrices with polynomial eigenvalue decay.

Findings

Improved bounds for matrix element errors using Chebyshev norm

Stronger bounds for symmetric positive definite matrices

Comparison showing bounds are tighter in practical cases

Abstract

Due to their importance in both data analysis and numerical algorithms, low rank approximations have recently been widely studied. They enable the handling of very large matrices. Tight error bounds for the computationally efficient Gaussian elimination based methods (skeleton approximations) are available. In practice, these bounds are useful for matrices with singular values which decrease quickly. Using the Chebyshev norm, this paper provides improved bounds for the errors of the matrix elements. These bounds are substantially better in the practically relevant cases where the eigenvalues decrease polynomially. Results are proven for general real rectangular matrices. Even stronger bounds are obtained for symmetric positive definite matrices. A simple example is given, comparing these new bounds to earlier ones.