Efficient Spectrum-Revealing CUR Matrix Decomposition

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Cheng Chen, Ming Gu, Zhihua Zhang, Weinan Zhang, Yong Yu ;
Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, PMLR 108:766-775, 2020.

Abstract

The CUR matrix decomposition is an important tool for low-rank matrix approximation. It approximates a data matrix though selecting a small number of columns and rows of the matrix. Those CUR algorithms with gap-dependent approximation bounds can obtain high approximation quality for matrices with good singular value spectrum decay, but they have impractically high time complexities. In this paper, we propose a novel CUR algorithm based on truncated LU factorization with an efficient variant of complete pivoting. Our algorithm has gap-dependent approximation bounds on both spectral and Frobenius norms while maintaining high efficiency. Numerical experiments demonstrate the effectiveness of our algorithm and verify our theoretical guarantees.

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