Computational Hardness of Certifying Bounds on Constrained PCA Problems

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 $\boldsymbol x$ in a constraint set $\mathcal{S} \subset \mathbb{R}^n$. For a certain class of normalized constraint sets $\mathcal{S}$ we show that, conditional on certain complexity-theoretic assumptions, there is no polynomial-time algorithm certifying a better upper bound than the largest eigenvalue of $\boldsymbol W$. A notable special case included in 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. 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 identifying computational thresholds was proposed in a sequence of recent works on the sum-of-squares hierarchy, and is believed 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 $\boldsymbol x \in \mathcal{S}$ is much larger than that of a GOE matrix.