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arXiv · 1011.0413

CUR from a Sparse Optimization Viewpoint

Jacob Bien·Ya Xu·Michael W. Mahoney
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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.

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BibTeXRIS

Jacob Bien, Ya Xu, Michael W. Mahoney. 2010-11-01. CUR from a Sparse Optimization Viewpoint. https://arxiv.org/abs/1011.0413

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