The Singular Value Decomposition, Applications and Beyond

TL;DR

This paper provides a comprehensive overview of the singular value decomposition (SVD), its mathematical foundations, and its applications in machine learning, including recent advances like randomized algorithms and low-rank matrix approximations.

Contribution

It offers a unified tutorial on SVD, explores unitarily invariant norms, and discusses recent developments in randomized algorithms for large-scale matrix computations.

Findings

Unified framework for SVD and matrix norms

Analysis of subdifferentials of unitarily invariant norms

Overview of recent randomized SVD techniques

Abstract

The singular value decomposition (SVD) is not only a classical theory in matrix computation and analysis, but also is a powerful tool in machine learning and modern data analysis. In this tutorial we first study the basic notion of SVD and then show the central role of SVD in matrices. Using majorization theory, we consider variational principles of singular values and eigenvalues. Built on SVD and a theory of symmetric gauge functions, we discuss unitarily invariant norms, which are then used to formulate general results for matrix low rank approximation. We study the subdifferentials of unitarily invariant norms. These results would be potentially useful in many machine learning problems such as matrix completion and matrix data classification. Finally, we discuss matrix low rank approximation and its recent developments such as randomized SVD, approximate matrix multiplication, CUR decomposition, and Nystrom approximation. Randomized algorithms are important approaches to large scale SVD as well as fast matrix computations.