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We prove a query complexity lower bound for approximating the top r dimensional eigenspace of a matrix. We consider an oracle model where, given a symmetric matrix M ∈ ℝd × d, an algorithm Alg is allowed to make T exact queries of the form w(i) = Mv(i) for i in {1,...,T}, where v(i) is drawn from a distribution which depends arbitrarily on the past queries and measurements {v(j),w(i)}1 ≤ j ≤ i−1. We show that for every gap ∈ (0,1/2], there exists a distribution over matrices M for which 1) gapr(M) = Ω(gap) (where gapr(M) is the normalized gap between the r and r+1-st largest-magnitude eigenvector of M), and 2) any Alg which takes fewer than const × r logd/√gap queries fails (with overwhelming probability) to identity a matrix V ∈ ℝd × r with orthonormal columns for which ⟨ V, MV⟩ ≥ (1 − const × gap)∑i=1r λi(M). Our bound requires only that d is a small polynomial in 1/gap and r, and matches the upper bounds of Musco and Musco ’15. Moreover, it establishes a strict separation between convex optimization and “strict-saddle” non-convex optimization of which PCA is a canonical example: in the former, first-order methods can have dimension-free iteration complexity, whereas in PCA, the iteration complexity of gradient-based methods must necessarily grow with the dimension. Our argument proceeds via a reduction to estimating a rank-r spike in a deformed Wigner model M =W + <pre>λ</pre> UU⊤, where W is from the Gaussian Orthogonal Ensemble, U is uniform on the d × r-Stieffel manifold and <pre>λ</pre> > 1 governs the size of the perturbation. Surprisingly, this ubiquitous random matrix model witnesses the worst-case rate for eigenspace approximation, and the ‘accelerated’ gap−1/2 in the rate follows as a consequence of the correspendence between the asymptotic eigengap and the size of the perturbation <pre>λ</pre>, when <pre>λ</pre> is near the “phase transition” <pre>λ</pre> = 1. To verify that d need only be polynomial in gap−1 and r, we prove a finite sample convergence theorem for top eigenvalues of a deformed Wigner matrix, which may be of independent interest. We then lower bound the above estimation problem with a novel technique based on Fano-style data-processing inequalities with truncated likelihoods; the technique generalizes the Bayes-risk lower bound of Chen et al. ’16, and we believe it is particularly suited to lower bounds in adaptive settings like the one considered in this paper.

Tight query complexity lower bounds for PCA via finite sample deformed wigner law