Computational Hardness of Certifying Bounds on Constrained PCA Problems.
arXiv: Data Structures and Algorithms(2020)
摘要
Given a random $n times n$ symmetric matrix $boldsymbol W$ drawn from the Gaussian orthogonal ensemble (GOE), we consider the problem of certifying an upper bound on the maximum value of the quadratic form $boldsymbol x^top boldsymbol W boldsymbol x$ over all vectors $boldsymbol x$ a constraint set $mathcal{S} subset mathbb{R}^n$. For a certain class of normalized constraint sets $mathcal{S}$, we give strong evidence that there is no polynomial-time algorithm certifying a better upper bound than the largest eigenvalue of $boldsymbol W$. A notable special case included our results is the hypercube $mathcal{S} = { pm 1 / sqrt{n}}^n$, which corresponds to the problem of certifying bounds on the Hamiltonian of the Sherrington-Kirkpatrick spin glass model from statistical physics. proof proceeds two steps. First, we give a reduction from the detection problem 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 $boldsymbol x in mathcal{S}$ is much larger than that of a GOE matrix.
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