Error Bounds for Random Matrix Approximation Schemes

TL;DR

This paper introduces new error bounds for random matrix approximation schemes across multiple norms, including the $( abla, 1)$ and spectral norms, enhancing understanding of their performance in matrix sparsification.

Contribution

It provides the first error bounds for approximation in the $( abla, 1)$ and $( abla, 2)$ norms, and analyzes spectral norm errors using Latala's result, broadening the theoretical framework.

Findings

Established optimality of $( abla, 1)$ and $( abla, 2)$ error bounds.

Derived concentration results for bounded entries in the three norms.

Predicted performance of existing sparsification schemes with competitive guarantees.

Abstract

Randomized matrix sparsification has proven to be a fruitful technique for producing faster algorithms in applications ranging from graph partitioning to semidefinite programming. In the decade or so of research into this technique, the focus has been--with few exceptions--on ensuring the quality of approximation in the spectral and Frobenius norms. For certain graph algorithms, however, the $(\infty,1)$ norm may be a more natural measure of performance. This paper addresses the problem of approximating a real matrix A by a sparse random matrix X with respect to several norms. It provides the first results on approximation error in the $(\infty, 1)$ and $(\infty, 2)$ norms, and it uses a result of Latala to study approximation error in the spectral norm. These bounds hold for random sparsification schemes which ensure that the entries of X are independent and average to the corresponding entries of A. Optimality of the $(\infty, 1)$ and $(\infty,2)$ error estimates is established. Concentration results for the three norms hold when the entries of X are uniformly bounded. The spectral error bound is used to predict the performance of several sparsification and quantization schemes that have appeared in the literature; the results are competitive with the performance guarantees given by earlier scheme-specific analyses.