Subspace-Orbit Randomized Decomposition for Low-rank Matrix Approximation

TL;DR

The paper introduces SOR-SVD, a new randomized matrix decomposition method that efficiently approximates large low-rank matrices with high accuracy, requiring minimal data passes and optimized for modern architectures.

Contribution

It proposes the novel Subspace-Orbit Randomized SVD algorithm, improving accuracy and efficiency over existing methods for low-rank matrix approximation.

Findings

Outperforms previous techniques in accuracy and efficiency.

Requires only a few passes through data, suitable for large matrices.

Can be optimized for advanced computer architectures.

Abstract

An efficient, accurate and reliable approximation of a matrix by one of lower rank is a fundamental task in numerical linear algebra and signal processing applications. In this paper, we introduce a new matrix decomposition approach termed Subspace-Orbit Randomized singular value decomposition (SOR-SVD), which makes use of random sampling techniques to give an approximation to a low-rank matrix. Given a large and dense data matrix of size $m\times n$ with numerical rank $k$, where $k \ll \text{min} \{m,n\}$, the algorithm requires a few passes through data, and can be computed in $O(mnk)$ floating-point operations. Moreover, the SOR-SVD algorithm can utilize advanced computer architectures, and, as a result, it can be optimized for maximum efficiency. The SOR-SVD algorithm is simple, accurate, and provably correct, and outperforms previously reported techniques in terms of accuracy and efficiency. Our numerical experiments support these claims.