Average-Case Integrality Gap for Non-Negative Principal Component Analysis
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Montanari and Richard (2015) asked whether a natural semidefinite programming (SDP) relaxation can effectively optimize 𝐱^⊤𝐖𝐱 over 𝐱 = 1 with x_i ≥ 0 for all coordinates i, where 𝐖∈ℝ^n × n is drawn from the Gaussian orthogonal ensemble (GOE) or a spiked matrix model. In small numerical experiments, this SDP appears to be tight for the GOE, producing a rank-one optimal matrix solution aligned with the optimal vector 𝐱. We prove, however, that as n →∞ the SDP is not tight, and certifies an upper bound asymptotically no better than the simple spectral bound λ_max(𝐖) on this objective function. We also provide evidence, using tools from recent literature on hypothesis testing with low-degree polynomials, that no subexponential-time certification algorithm can improve on this behavior. Finally, we present further numerical experiments estimating how large n would need to be before this limiting behavior becomes evident, providing a cautionary example against extrapolating asymptotics of SDPs in high dimension from their efficacy in small "laptop scale" computations.
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