Do algorithms and barriers for sparse principal component analysis extend to other structured settings?

Guanyi Wang; Mengqi Lou; Ashwin Pananjady

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arXiv · 2307.13535

Do algorithms and barriers for sparse principal component analysis extend to other structured settings?

Guanyi Wang·Mengqi Lou·Ashwin Pananjady

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Abstract

We study a principal component analysis problem under the spiked Wishart model in which the structure in the signal is captured by a class of union-of-subspace models. This general class includes vanilla sparse PCA as well as its variants with graph sparsity. With the goal of studying these problems under a unified statistical and computational lens, we establish fundamental limits that depend on the geometry of the problem instance, and show that a natural projected power method exhibits local convergence to the statistically near-optimal neighborhood of the solution. We complement these results with end-to-end analyses of two important special cases given by path and tree sparsity in a general basis, showing initialization methods and matching evidence of computational hardness. Overall, our results indicate that several of the phenomena observed for vanilla sparse PCA extend in a natural fashion to its structured counterparts.

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Guanyi Wang, Mengqi Lou, Ashwin Pananjady. 2023-07-25. Do algorithms and barriers for sparse principal component analysis extend to other structured settings?. https://arxiv.org/abs/2307.13535

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