A random algorithm for low-rank decomposition of large-scale matrices with missing entries

TL;DR

This paper introduces a fast, memory-efficient random submatrix method (RSM) for low-rank matrix decomposition, effective even with missing data, outperforming existing algorithms in speed and resource usage.

Contribution

The paper presents a novel RSM algorithm that efficiently computes low-rank decompositions of large matrices with missing entries, with proven theoretical guarantees and superior performance.

Findings

RSM is significantly faster than state-of-the-art algorithms.

RSM requires less memory while maintaining high accuracy.

Experimental results show RSM's effectiveness on large-scale and real datasets.

Abstract

A Random SubMatrix method (RSM) is proposed to calculate the low-rank decomposition of large-scale matrices with known entry percentage \rho. RSM is very fast as the floating-point operations (flops) required are compared favorably with the state-of-the-art algorithms. Meanwhile RSM is very memory-saving. With known entries homogeneously distributed in the given matrix, sub-matrices formed by known entries are randomly selected. According to the just proved theorem that subspace related to smaller singular values is less perturbed by noise, the null vectors or the right singular vectors associated with the minor singular values are calculated for each submatrix. The vectors are the null vectors of the corresponding submatrix in the ground truth of the given large-scale matrix. If enough sub-matrices are randomly chosen, the low-rank decomposition is estimated. The experimental results on random synthetical matrices with sizes such as 131072X1024 and on real data sets indicate that RSM is much faster and memory-saving, and, meanwhile, has considerable high precision achieving or approximating to the best.