Uniform Sampling for Matrix Approximation

TL;DR

This paper investigates the effectiveness of uniform row sampling for matrix approximation, demonstrating it preserves enough information for improved approximations and introducing efficient algorithms that maintain sparsity.

Contribution

It provides new insights into uniform sampling's capabilities, showing it can be used effectively for matrix approximation and introduces input-sparsity time algorithms that reweight rows to reduce coherence.

Findings

Uniform sampling preserves a large fraction of the original matrix's information.

Reweighting a small subset of rows can make any matrix have low coherence.

New algorithms run in input-sparsity time and maintain row sparsity.

Abstract

Random sampling has become a critical tool in solving massive matrix problems. For linear regression, a small, manageable set of data rows can be randomly selected to approximate a tall, skinny data matrix, improving processing time significantly. For theoretical performance guarantees, each row must be sampled with probability proportional to its statistical leverage score. Unfortunately, leverage scores are difficult to compute. A simple alternative is to sample rows uniformly at random. While this often works, uniform sampling will eliminate critical row information for many natural instances. We take a fresh look at uniform sampling by examining what information it does preserve. Specifically, we show that uniform sampling yields a matrix that, in some sense, well approximates a large fraction of the original. While this weak form of approximation is not enough for solving linear regression directly, it is enough to compute a better approximation. This observation leads to simple iterative row sampling algorithms for matrix approximation that run in input-sparsity time and preserve row structure and sparsity at all intermediate steps. In addition to an improved understanding of uniform sampling, our main proof introduces a structural result of independent interest: we show that every matrix can be made to have low coherence by reweighting a small subset of its rows.