CUR Algorithm for Partially Observed Matrices

TL;DR

This paper introduces a CUR matrix decomposition algorithm for partially observed matrices, enabling low-rank approximation using sampled rows, columns, and entries, with theoretical guarantees and improved sample complexity.

Contribution

It develops a novel CUR decomposition method that works with partial observations, reducing the need for full matrix access and improving sample efficiency for matrix recovery.

Findings

Achieves spectral norm error bounds for full-rank matrices.

Requires only O(n r log r) observed entries for perfect rank r matrix recovery.

Empirical results confirm theoretical advantages on synthetic and real data.

Abstract

CUR matrix decomposition computes the low rank approximation of a given matrix by using the actual rows and columns of the matrix. It has been a very useful tool for handling large matrices. One limitation with the existing algorithms for CUR matrix decomposition is that they need an access to the {\it full} matrix, a requirement that can be difficult to fulfill in many real world applications. In this work, we alleviate this limitation by developing a CUR decomposition algorithm for partially observed matrices. In particular, the proposed algorithm computes the low rank approximation of the target matrix based on (i) the randomly sampled rows and columns, and (ii) a subset of observed entries that are randomly sampled from the matrix. Our analysis shows the relative error bound, measured by spectral norm, for the proposed algorithm when the target matrix is of full rank. We also show that only $O(n r\ln r)$ observed entries are needed by the proposed algorithm to perfectly recover a rank $r$ matrix of size $n\times n$, which improves the sample complexity of the existing algorithms for matrix completion. Empirical studies on both synthetic and real-world datasets verify our theoretical claims and demonstrate the effectiveness of the proposed algorithm.