Computational Hardness of Certifying Bounds on Constrained PCA Problems
share
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⊂R^n. For a certain class of normalized constraint sets S, we give strong evidence that there is no polynomial-time algorithm certifying 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 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, using a method of Hopkins and Steurer based on approximating the likelihood ratio with a low-degree polynomial. 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.
READ FULL TEXT