Literature survey on low rank approximation of matrices

TL;DR

This survey reviews various low rank matrix approximation techniques, comparing classical deterministic methods with more recent randomized and cross approximation methods, highlighting their computational complexities and applications.

Contribution

It provides a comprehensive overview of existing low rank approximation algorithms, including classical, randomized, and cross approximation techniques, with extensive references.

Findings

Classical methods are computationally expensive for large matrices.

Randomized algorithms offer less costly alternatives but are not linear in complexity.

Cross/Skeleton approximation achieves linear complexity in matrix dimension.

Abstract

Low rank approximation of matrices has been well studied in literature. Singular value decomposition, QR decomposition with column pivoting, rank revealing QR factorization (RRQR), Interpolative decomposition etc are classical deterministic algorithms for low rank approximation. But these techniques are very expensive $(O(n^{3})$ operations are required for $n\times n$ matrices). There are several randomized algorithms available in the literature which are not so expensive as the classical techniques (but the complexity is not linear in n). So, it is very expensive to construct the low rank approximation of a matrix if the dimension of the matrix is very large. There are alternative techniques like Cross/Skeleton approximation which gives the low-rank approximation with linear complexity in n . In this article we review low rank approximation techniques briefly and give extensive references of many techniques.