  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.
