Accuracy and Stability of CUR decompositions with Oversampling

TL;DR

This paper analyzes the accuracy and stability of CUR matrix decompositions with oversampling, demonstrating improved numerical stability and accuracy through theoretical insights and an algorithmic approach.

Contribution

It introduces a stable implementation of CUR decompositions with oversampling and proposes a new algorithm inspired by the theory of CUR and cosine-sine decomposition.

Findings

Oversampling improves CUR decomposition accuracy.

Oversampling enhances numerical stability of CUR.

The proposed algorithm is competitive in experiments.

Abstract

This work investigates the accuracy and numerical stability of CUR decompositions with oversampling. The CUR decomposition approximates a matrix using a subset of columns and rows of the matrix. When the number of columns and the rows are the same, the CUR decomposition can become unstable and less accurate due to the presence of the matrix inverse in the core matrix. Nevertheless, we demonstrate that the CUR decomposition can be implemented in a numerical stable manner and illustrate that oversampling, which increases either the number of columns or rows in the CUR decomposition, can enhance its accuracy and stability. Additionally, this work devises an algorithm for oversampling motivated by the theory of the CUR decomposition and the cosine-sine decomposition, whose competitiveness is illustrated through experiments.