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DOI: 10.4230/LIPIcs.ITCS.2020.78
URN: urn:nbn:de:0030-drops-117633
URL: https://drops.dagstuhl.de/opus/volltexte/2020/11763/
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Bandeira, Afonso S. ; Kunisky, Dmitriy ; Wein, Alexander S.

Computational Hardness of Certifying Bounds on Constrained PCA Problems

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Abstract

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.

BibTeX - Entry

@InProceedings{bandeira_et_al:LIPIcs:2020:11763,
  author =	{Afonso S. Bandeira and Dmitriy Kunisky and Alexander S. Wein},
  title =	{{Computational Hardness of Certifying Bounds on Constrained PCA Problems}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{78:1--78:29},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Thomas Vidick},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/11763},
  URN =		{urn:nbn:de:0030-drops-117633},
  doi =		{10.4230/LIPIcs.ITCS.2020.78},
  annote =	{Keywords: Certification, Sherrington-Kirkpatrick model, spiked Wishart model, low-degree likelihood ratio}
}

Keywords: Certification, Sherrington-Kirkpatrick model, spiked Wishart model, low-degree likelihood ratio
Seminar: 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)
Issue Date: 2020
Date of publication: 10.01.2020


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