Compressed Randomized UTV Decompositions for Low-Rank Approximations and Big Data Applications

TL;DR

This paper presents CoR-UTV, a new randomized matrix decomposition method that efficiently computes low-rank approximations with high accuracy, suitable for large-scale data and modern computational platforms.

Contribution

Introduction of CoR-UTV, a novel randomized low-rank matrix decomposition algorithm with a power method variant, optimized for efficiency and accuracy in big data applications.

Findings

CoR-UTV requires few data passes and runs in O(mnk) operations.

CoR-UTV outperforms existing methods in efficiency and accuracy.

Validated on synthetic and real data for image reconstruction and PCA.

Abstract

Low-rank matrix approximations play a fundamental role in numerical linear algebra and signal processing applications. This paper introduces a novel rank-revealing matrix decomposition algorithm termed Compressed Randomized UTV (CoR-UTV) decomposition along with a CoR-UTV variant aided by the power method technique. CoR-UTV is primarily developed to compute an approximation to a low-rank input matrix by making use of random sampling schemes. Given a large and dense matrix of size $m\times n$ with numerical rank $k$, where $k \ll \text{min} \{m,n\}$, CoR-UTV requires a few passes over the data, and runs in $O(mnk)$ floating-point operations. Furthermore, CoR-UTV can exploit modern computational platforms and, consequently, can be optimized for maximum efficiency. CoR-UTV is simple and accurate, and outperforms reported alternative methods in terms of efficiency and accuracy. Simulations with synthetic data as well as real data in image reconstruction and robust principal component analysis applications support our claims.