A general error analysis for randomized low-rank approximation methods

TL;DR

This paper develops a comprehensive error analysis framework for randomized low-rank approximation methods, providing deterministic and probabilistic bounds that improve upon existing results and are validated through numerical experiments.

Contribution

It introduces a unified error analysis for spectral and Frobenius norms, extending and refining previous bounds, including for the Randomized SVD, under minimal assumptions.

Findings

Derived deterministic error bounds with minimal assumptions

Established expectation-based error bounds for Gaussian cases

Numerical experiments confirm the tightness of the bounds

Abstract

We propose a general error analysis related to the low-rank approximation of a given real matrix in both the spectral and Frobenius norms. First, we derive deterministic error bounds that hold with some minimal assumptions. Second, we derive error bounds in expectation in the non-standard Gaussian case, assuming a non-trivial mean and a general covariance matrix for the random matrix variable. The proposed analysis generalizes and improves the error bounds for spectral and Frobenius norms proposed by Halko, Martinsson and Tropp. Third, we consider the Randomized Singular Value Decomposition and specialize our error bounds in expectation in this setting. Numerical experiments on an instructional synthetic test case demonstrate the tightness of the new error bounds.