A Practical Guide to Randomized Matrix Computations with MATLAB Implementations

TL;DR

This paper provides an accessible, practical guide to randomized matrix algorithms, focusing on intuition, derivation, and MATLAB implementations to improve scalability of large matrix computations.

Contribution

It offers a user-friendly, implementation-focused overview of randomized matrix algorithms, making advanced techniques accessible to practitioners without deep mathematical background.

Findings

Algorithms are summarized in MATLAB code snippets.

Focus on intuition and implementation rather than deep mathematical theory.

Enhances scalability of large matrix operations in real-world applications.

Abstract

Matrix operations such as matrix inversion, eigenvalue decomposition, singular value decomposition are ubiquitous in real-world applications. Unfortunately, many of these matrix operations so time and memory expensive that they are prohibitive when the scale of data is large. In real-world applications, since the data themselves are noisy, machine-precision matrix operations are not necessary at all, and one can sacrifice a reasonable amount of accuracy for computational efficiency. In recent years, a bunch of randomized algorithms have been devised to make matrix computations more scalable. Mahoney (2011) and Woodruff (2014) have written excellent but very technical reviews of the randomized algorithms. Differently, the focus of this manuscript is on intuition, algorithm derivation, and implementation. This manuscript should be accessible to people with knowledge in elementary matrix algebra but unfamiliar with randomized matrix computations. The algorithms introduced in this manuscript are all summarized in a user-friendly way, and they can be implemented in lines of MATLAB code. The readers can easily follow the implementations even if they do not understand the maths and algorithms.