CUR Algorithm with Incomplete Matrix Observation

TL;DR

This paper introduces a CUR matrix approximation algorithm that operates effectively with incomplete data, using sampled rows, columns, and entries, and guarantees accurate recovery even for non-low-rank matrices.

Contribution

The authors develop a CUR-based low rank approximation method that works with partial observations and does not require the target matrix to be low rank, simplifying computation.

Findings

Achieves perfect recovery with O(rn log n) observed entries.

Replaces trace norm regularization with standard regression.

Provides strong guarantees for non-low-rank matrices.

Abstract

CUR matrix decomposition is a randomized algorithm that can efficiently compute the low rank approximation for a given rectangle matrix. One limitation with the existing CUR algorithms is that they require an access to the full matrix A for computing U. In this work, we aim to alleviate this limitation. In particular, we assume that besides having an access to randomly sampled d rows and d columns from A, we only observe a subset of randomly sampled entries from A. Our goal is to develop a low rank approximation algorithm, similar to CUR, based on (i) randomly sampled rows and columns from A, and (ii) randomly sampled entries from A. The proposed algorithm is able to perfectly recover the target matrix A with only O(rn log n) number of observed entries. In addition, instead of having to solve an optimization problem involved trace norm regularization, the proposed algorithm only needs to solve a standard regression problem. Finally, unlike most matrix completion theories that hold only when the target matrix is of low rank, we show a strong guarantee for the proposed algorithm even when the target matrix is not low rank.