Bayesian Matrix Decomposition and Applications

TL;DR

This book provides a comprehensive, self-contained introduction to Bayesian matrix decomposition techniques, covering their mathematical foundations, significance, and applications, with rigorous proofs and minimal prerequisites.

Contribution

It offers a detailed overview of key Bayesian matrix decomposition methods and their applications, filling a gap in accessible, rigorous educational resources.

Findings

Summarizes important Bayesian matrix decomposition methods.

Highlights applications in various fields.

Provides rigorous proofs and mathematical foundations.

Abstract

The sole aim of this book is to give a self-contained introduction to concepts and mathematical tools in Bayesian matrix decomposition 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 Bayesian matrix decomposition and given the paucity of scope to present this discussion, e.g., the separated analysis of variational inference for conducting the optimization. We refer the reader to literature in the field of Bayesian analysis for a more detailed introduction to the related fields. This book is primarily a summary of purpose, significance of important Bayesian matrix decomposition methods, e.g., real-valued decomposition, nonnegative matrix factorization, Bayesian interpolative decomposition, and the origin and complexity of the methods which shed light on their applications. The mathematical prerequisite is a first course in statistics and linear algebra. Other than this modest background, the development is self-contained, with rigorous proof provided throughout.