CUR from a Sparse Optimization Viewpoint

TL;DR

This paper analyzes the CUR matrix decomposition from a sparse optimization perspective, revealing its implicit sparse regression formulation and proposing a new sparse PCA method that mimics CUR's sparsity structure.

Contribution

It demonstrates that CUR implicitly optimizes a sparse regression objective and introduces a sparse PCA method inspired by CUR's sparsity pattern.

Findings

CUR is implicitly optimizing a sparse regression objective.

CUR's sparsity has a unique structure.

A new sparse PCA method mimicking CUR's sparsity is proposed.

Abstract

The CUR decomposition provides an approximation of a matrix $X$ that has low reconstruction error and that is sparse in the sense that the resulting approximation lies in the span of only a few columns of $X$. In this regard, it appears to be similar to many sparse PCA methods. However, CUR takes a randomized algorithmic approach, whereas most sparse PCA methods are framed as convex optimization problems. In this paper, we try to understand CUR from a sparse optimization viewpoint. We show that CUR is implicitly optimizing a sparse regression objective and, furthermore, cannot be directly cast as a sparse PCA method. We also observe that the sparsity attained by CUR possesses an interesting structure, which leads us to formulate a sparse PCA method that achieves a CUR-like sparsity.