Sketching as a Tool for Numerical Linear Algebra

TL;DR

This survey reviews recent developments in linear sketching techniques that enable faster algorithms for numerical linear algebra problems like least squares, low-rank approximation, and graph sparsification by compressing matrices.

Contribution

It provides a comprehensive overview of how linear sketching is applied to various numerical linear algebra problems and discusses its advantages and limitations.

Findings

Sketching accelerates computations for large matrices.

Effective for least squares, low-rank approximation, and graph sparsification.

Discusses limitations and variants of sketching methods.

Abstract

This survey highlights the recent advances in algorithms for numerical linear algebra that have come from the technique of linear sketching, whereby given a matrix, one first compresses it to a much smaller matrix by multiplying it by a (usually) random matrix with certain properties. Much of the expensive computation can then be performed on the smaller matrix, thereby accelerating the solution for the original problem. In this survey we consider least squares as well as robust regression problems, low rank approximation, and graph sparsification. We also discuss a number of variants of these problems. Finally, we discuss the limitations of sketching methods.