Error Estimation for Sketched SVD via the Bootstrap

TL;DR

This paper introduces a bootstrap-based, data-driven error estimation method for sketched SVDs that enables adaptive accuracy assessment without full matrix passes, supported by theory and experiments.

Contribution

It develops a computationally efficient bootstrap approach for estimating errors in sketched SVDs, allowing adaptive accuracy control based on data.

Findings

The method provides reliable error estimates for sketched SVDs.

It requires no additional passes over the full matrix, saving computational resources.

Experimental results confirm the effectiveness and theoretical guarantees of the approach.

Abstract

In order to compute fast approximations to the singular value decompositions (SVD) of very large matrices, randomized sketching algorithms have become a leading approach. However, a key practical difficulty of sketching an SVD is that the user does not know how far the sketched singular vectors/values are from the exact ones. Indeed, the user may be forced to rely on analytical worst-case error bounds, which do not account for the unique structure of a given problem. As a result, the lack of tools for error estimation often leads to much more computation than is really necessary. To overcome these challenges, this paper develops a fully data-driven bootstrap method that numerically estimates the actual error of sketched singular vectors/values. In particular, this allows the user to inspect the quality of a rough initial sketched SVD, and then adaptively predict how much extra work is needed to reach a given error tolerance. Furthermore, the method is computationally inexpensive, because it operates only on sketched objects, and it requires no passes over the full matrix being factored. Lastly, the method is supported by theoretical guarantees and a very encouraging set of experimental results.