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Computational Hardness of Certifying Bounds on Constrained PCA Problems

Authors Afonso S. Bandeira, Dmitriy Kunisky, Alexander S. Wein

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Afonso S. Bandeira
  • Dept. of Mathematics, ETH Zurich, Switzerland
Dmitriy Kunisky
  • Dept. of Mathematics, Courant Institute of Mathematical Sciences, New York University, USA
Alexander S. Wein
  • Dept. of Mathematics, Courant Institute of Mathematical Sciences, New York University, USA


We thank Andrea Montanari and Samuel B. Hopkins for insightful discussions.

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Afonso S. Bandeira, Dmitriy Kunisky, and Alexander S. Wein. Computational Hardness of Certifying Bounds on Constrained PCA Problems. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 78:1-78:29, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
  • Certification
  • Sherrington-Kirkpatrick model
  • spiked Wishart model
  • low-degree likelihood ratio


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