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Computational Hardness of Certifying Bounds on Constrained PCA Problems Bandeira, Afonso S. Kunisky, Dmitriy Wein, Alexander S. Certification Sherrington-Kirkpatrick model spiked Wishart model low-degree likelihood ratio Given a random n × n symmetric matrix ? drawn from the Gaussian orthogonal ensemble (GOE), we consider the problem of certifying an upper bound on the maximum value of the quadratic form ?^⊤ ? ? over all vectors ? in a constraint set ? ⊂ ℝⁿ. For a certain class of normalized constraint sets we show that, conditional on a certain complexity-theoretic conjecture, no polynomial-time algorithm can certify a better upper bound than the largest eigenvalue of ?. A notable special case included in our results is the hypercube ? = {±1/√n}ⁿ, which corresponds to the problem of certifying bounds on the Hamiltonian of the Sherrington-Kirkpatrick spin glass model from statistical physics. Our results suggest a striking gap between optimization and certification for this problem. Our proof proceeds in two steps. First, we give a reduction from the detection problem in the negatively-spiked Wishart model to the above certification problem. We then give evidence that this Wishart detection problem is computationally hard below the classical spectral threshold, by showing that no low-degree polynomial can (in expectation) distinguish the spiked and unspiked models. This method for predicting computational thresholds was proposed in a sequence of recent works on the sum-of-squares hierarchy, and is conjectured to be correct for a large class of problems. Our proof can be seen as constructing a distribution over symmetric matrices that appears computationally indistinguishable from the GOE, yet is supported on matrices whose maximum quadratic form over ? ∈ ? is much larger than that of a GOE matrix. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Afonso S. Bandeira and Dmitriy Kunisky and Alexander S. Wein 2020 Is Part Of LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ITCS.2020.78 urn:nbn:de:0030-drops-117633 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.78 eng https://creativecommons.org/licenses/by/3.0/legalcode