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

TL;DR

This paper investigates structured sparse PCA problems within a union-of-subspace model, establishing fundamental limits and analyzing algorithmic convergence, thereby extending insights from vanilla sparse PCA to more complex structured settings.

Contribution

It introduces a unified framework for structured sparse PCA, derives fundamental limits, and analyzes the convergence of algorithms in these settings.

Findings

Projected power method converges locally to near-optimal solutions.

Fundamental limits depend on the geometry of the problem.

Structured sparse PCA phenomena extend from vanilla sparse PCA.

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.