Abstract
Given a random n × n symmetric matrix 𝑾 drawn from the Gaussian orthogonal ensemble (GOE), we consider the problem of certifying an upper bound on the maximum value of the quadratic form 𝒙^⊤ 𝑾 𝒙 over all vectors 𝒙 in a constraint set 𝒮 ⊂ ℝⁿ. For a certain class of normalized constraint sets we show that, conditional on a certain complexitytheoretic conjecture, no polynomialtime algorithm can certify a better upper bound than the largest eigenvalue of 𝑾. A notable special case included in our results is the hypercube 𝒮 = {±1/√n}ⁿ, which corresponds to the problem of certifying bounds on the Hamiltonian of the SherringtonKirkpatrick 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 negativelyspiked 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 lowdegree 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 sumofsquares 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 𝒙 ∈ 𝒮 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 = {78:178:29},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771344},
ISSN = {18688969},
year = {2020},
volume = {151},
editor = {Thomas Vidick},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/11763},
URN = {urn:nbn:de:0030drops117633},
doi = {10.4230/LIPIcs.ITCS.2020.78},
annote = {Keywords: Certification, SherringtonKirkpatrick model, spiked Wishart model, lowdegree likelihood ratio}
}
Keywords: 

Certification, SherringtonKirkpatrick model, spiked Wishart model, lowdegree likelihood ratio 
Collection: 

11th Innovations in Theoretical Computer Science Conference (ITCS 2020) 
Issue Date: 

2020 
Date of publication: 

06.01.2020 