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Title: Computational Hardness of Certifying Bounds on Constrained PCA Problems
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 complexity-theoretic conjecture, no polynomial-time 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 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 𝒙 ∈ 𝒮 is much larger than that of a GOE matrix.  more » « less
Award ID(s):
1712730 1719545
PAR ID:
10164438
Author(s) / Creator(s):
; ;
Date Published:
Journal Name:
11th Innovations in Theoretical Computer Science Conference (ITCS 2020)
Volume:
151
Page Range / eLocation ID:
78:1 - 78:29
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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