Row-aware Randomized SVD with applications

TL;DR

This paper introduces a row-aware modification to randomized SVD that improves approximation quality and efficiency, especially for matrices with significant singular value gaps, supported by theoretical bounds and numerical experiments.

Contribution

It proposes a novel row-aware randomized SVD method with enhanced accuracy and efficiency, including a subsampling variant, supported by new error bounds and empirical validation.

Findings

Improved approximation of the column space of matrices with singular value gaps.

A subsampling variant increases efficiency while maintaining competitive accuracy.

Theoretical error bounds outperform existing results for certain matrices.

Abstract

The randomized singular value decomposition proposed in [27] has certainly become one of the most well-established randomization-based algorithms in numerical linear algebra. The key ingredient of the entire procedure is the computation of a subspace which is close to the column space of the target matrix $\mathbf{A}$ up to a certain probabilistic confidence. In this paper we employ a modification to the standard randomized SVD procedure which leads, in general, to better approximations to $\text{Range}(\mathbf{A})$ at the same computational cost. To this end, we explicitly construct information from the row space of $\mathbf{A}$ enhancing the quality of the approximation. We derive novel error bounds which improve over existing results for $\mathbf{A}$ having important gaps in its singular values. We also observe that very few pieces of information from $\text{Range}(\mathbf{A}^T)$ may be necessary. We thus design a variant of this algorithm equipped with a subsampling step which largely increases the efficiency of the procedure while often attaining competitive accuracy records. Our findings are supported by both theoretical analysis and numerical results.