The Singular Value Decomposition, Applications and Beyond
It provides a comprehensive overview of SVD for researchers and practitioners in machine learning and data analysis, but is incremental as it synthesizes existing knowledge without new results.
This tutorial reviews the singular value decomposition (SVD) as a fundamental tool in machine learning and data analysis, covering its theory, applications in matrix low-rank approximation, and recent developments like randomized SVD and CUR decomposition.
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.