Numerical Matrix Decomposition

TL;DR

This survey provides a comprehensive introduction to matrix decomposition techniques, their mathematical foundations, and applications, emphasizing their importance in numerical linear algebra and machine learning.

Contribution

It offers a self-contained overview of key matrix decomposition methods and discusses their theoretical and practical aspects, serving as an accessible entry point for further study.

Findings

Overview of fundamental matrix decomposition techniques

Discussion of applications in machine learning and numerical analysis

Highlighting computational considerations like floating point operations

Abstract

In 1954, Alston S. Householder published \textit{Principles of Numerical Analysis}, one of the first modern treatments on matrix decomposition that favored a (block) LU decomposition-the factorization of a matrix into the product of lower and upper triangular matrices. And now, matrix decomposition has become a core technology in machine learning, largely due to the development of the back propagation algorithm in fitting a neural network. The sole aim of this survey is to give a self-contained introduction to concepts and mathematical tools in numerical linear algebra and matrix analysis in order to seamlessly introduce matrix decomposition techniques and their applications in subsequent sections. However, we clearly realize our inability to cover all the useful and interesting results concerning matrix decomposition and given the paucity of scope to present this discussion, e.g., the separated analysis of the Euclidean space, Hermitian space, Hilbert space, and things in the complex domain. We refer the reader to literature in the field of linear algebra for a more detailed introduction to the related fields. Keywords: Existence and computing of matrix decompositions, Floating point operations (flops), Low-rank approximation, Pivot, LU/PLU decomposition, CR/CUR/Skeleton decomposition, Coordinate transformation, ULV/URV decomposition, Rank decomposition, Rank revealing decomposition, Update/downdate, Tensor decomposition.