Matrix Decompositions using sub-Gaussian Random Matrices

TL;DR

This paper introduces a new matrix decomposition algorithm using sub-Gaussian random matrices that is faster in practice and achieves comparable error rates to existing methods, with a novel error bound independent of the top singular values.

Contribution

The paper demonstrates that sub-Gaussian matrices are metric conserving and presents a new efficient algorithm for low-rank matrix approximation with a novel error bound.

Findings

Algorithm achieves similar error rates as state-of-the-art methods.

Proposed method is faster in practice.

Error bound is independent of the first r singular values.

Abstract

In recent years, several algorithms, which approximate matrix decomposition, have been developed. These algorithms are based on metric conservation features for linear spaces of random projection types. We show that an i.i.d sub-Gaussian matrix with large probability to have zero entries is metric conserving. We also present a new algorithm, which achieves with high probability, a rank $r$ decomposition approximation for an $m \times n$ matrix that has an asymptotic complexity like state-of-the-art algorithms. We derive an error bound that does not depend on the first $r$ singular values. Although the proven error bound is not as tight as the state-of-the-art bound, experiments show that the proposed algorithm is faster in practice, while getting the same error rates as the state-of-the-art algorithms get.