Improved Low-rank Matrix Decompositions via the Subsampled Randomized Hadamard Transform

TL;DR

This paper analyzes two randomized algorithms for low-rank matrix decompositions using the Subsampled Randomized Hadamard Transform, improving approximation bounds and running times compared to previous versions.

Contribution

It provides a novel analysis that enhances approximation bounds and modifies existing algorithms to significantly reduce running time.

Findings

Improved approximation bounds for both algorithms.

Modified second algorithm with faster running time.

Enhanced theoretical understanding of randomized low-rank decompositions.

Abstract

We comment on two randomized algorithms for constructing low-rank matrix decompositions. Both algorithms employ the Subsampled Randomized Hadamard Transform [14]. The first algorithm appeared recently in [9]; here, we provide a novel analysis that significantly improves the approximation bound obtained in [9]. A preliminary version of the second algorithm appeared in [7]; here, we present a mild modification of this algorithm that achieves the same approximation bound but significantly improves the corresponding running time.