Accelerating Randomized Algorithms for Low-Rank Matrix Approximation

TL;DR

This paper introduces new randomized algorithms using sparse and Bernoulli matrices to accelerate low-rank matrix approximation, maintaining accuracy while reducing computational costs compared to traditional Gaussian-based methods.

Contribution

The paper proposes three novel randomized algorithms replacing Gaussian matrices with sparse and Bernoulli matrices in exttt{farPCA}, achieving faster computation with comparable accuracy.

Findings

Algorithms achieve similar accuracy to exttt{farPCA}

Reduced computational cost in matrix multiplication

Tighter error bounds derived using random matrix theory

Abstract

Randomized algorithms are overwhelming methods for low-rank approximation that can alleviate the computational expenditure with great reliability compared to deterministic algorithms. A crucial thought is generating a standard Gaussian matrix $\mathbf{G}$ and subsequently obtaining the orthonormal basis of the range of $\mathbf{AG}$ for a given matrix $\mathbf{A}$. Recently, the \texttt{farPCA} algorithm offers a framework for randomized algorithms, but the dense Gaussian matrix remains computationally expensive. Motivated by this, we introduce the standardized Bernoulli, sparse sign, and sparse Gaussian matrices to replace the standard Gaussian matrix in \texttt{farPCA} for accelerating computation. These three matrices possess a low computational expenditure in matrix-matrix multiplication and converge in distribution to a standard Gaussian matrix when multiplied by an orthogonal matrix under a mild condition. Therefore, the three corresponding proposed algorithms can serve as a superior alternative to fast adaptive randomized PCA (\texttt{farPCA}). Finally, we leverage random matrix theory (RMT) to derive a tighter error bound for \texttt{farPCA} without shifted techniques. Additionally, we extend this improved error bound to the error analysis of our three fast algorithms, ensuring that the proposed methods deliver more accurate approximations for large-scale matrices. Numerical experiments validate that the three algorithms achieve asymptotically the same performance as \texttt{farPCA} but with lower costs, offering a more efficient approach to low-rank matrix approximation.