Randomized Algorithms for Low-Rank Matrix and Tensor Decompositions

TL;DR

This survey reviews recent randomized algorithms for efficient low-rank matrix and tensor decompositions, highlighting advances in matrix sketching, sampling, and their extensions to tensor formats like CP and Tucker.

Contribution

It provides a comprehensive overview of recent developments in randomized algorithms for low-rank matrix and tensor decompositions, including new sketching and sampling techniques.

Findings

Randomized algorithms accelerate classical low-rank matrix decompositions.

Recent advances include fast matrix sketching and sampling methods.

Extensions to tensor decompositions like CP and Tucker are discussed.

Abstract

This paper surveys randomized algorithms in numerical linear algebra for low-rank decompositions of matrices and tensors. The survey begins with a review of classical matrix algorithms that can be accelerated by randomized dimensionality reduction, such as the singular value decomposition (SVD) or interpolative (ID) and CUR decompositions. Recent advances in randomized dimensionality reduction are discussed, including new methods of fast matrix sketching and sampling techniques, which are incorporated into classical matrix algorithms for fast low-rank matrix approximations. The extension of randomized matrix algorithms to tensors is then explored for several low-rank tensor decompositions in the CP and Tucker formats, including the higher-order SVD, ID, and CUR decomposition.