CUR from a Sparse Optimization Viewpoint

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.

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Jacob Bien, Ya Xu, Michael W. Mahoney
• Keywords: CORR 2010, Education, Pca Method, Sparse, Sparse Pca Method