Randomized subspace iteration: Analysis of canonical angles and unitarily invariant norms

TL;DR

This paper analyzes the randomized subspace iteration method for low-rank matrix approximation, providing bounds on canonical angles, unitarily invariant norms, and singular values, applicable to various initial guesses.

Contribution

It introduces new bounds for the accuracy of randomized subspace iteration in terms of canonical angles, unitarily invariant norms, and singular values, generalizing previous results.

Findings

Bounds are effective for Gaussian initial guesses.

Results apply to any starting guess with minimal assumptions.

Numerical experiments confirm the bounds' accuracy.

Abstract

This paper is concerned with the analysis of the randomized subspace iteration for the computation of low-rank approximations. We present three different kinds of bounds. First, we derive both bounds for the canonical angles between the exact and the approximate singular subspaces. Second, we derive bounds for the low-rank approximation in any unitarily invariant norm (including the Schatten-p norm). This generalizes the bounds for Spectral and Frobenius norms found in the literature. Third, we present bounds for the accuracy of the singular values. The bounds are structural in that they are applicable to any starting guess, be it random or deterministic, that satisfies some minimal assumptions. Specialized bounds are provided when a Gaussian random matrix is used as the starting guess. Numerical experiments demonstrate the effectiveness of the proposed bounds.