Sum-of-Squares Lower Bounds for Sherrington-Kirkpatrick via Planted Affine Planes

The Sum-of-Squares (SoS) hierarchy is a semi-definite programming meta-algorithm that captures state-of-the-art polynomial time guarantees for many optimization problems such as Max-k-CSPs and Tensor PCA. On the flip side, a SoS lower bound provides evidence of hardness, which is particularly relevant to average-case problems for which NPhardness may not be available. In this paper, we consider the following average case problem, which we call the Planted Affine Planes (PAP) problem: Given m random vectors d(1),..., d(m) in Rn, can we prove that there is no vector v. Rn such that for all u is an element of [m], < v, du >(2) = 1? In other words, can we prove that m random vectors are not all contained in two parallel hyperplanes at equal distance from the origin? We prove that for m <= n(3/2-e), with high probability, degree-nO(e) SoS fails to refute the existence of such a vector v. When the vectors d(1),..., d(m) are chosen from the multivariate normal distribution, the PAP problem is equivalent to the problem of proving that a random n-dimensional subspace of Rm does not contain a boolean vector. As shown by Mohanty-RaghavendraXu [STOC 2020], a lower bound for this problem implies a lower bound for the problem of certifying energy upper bounds on the Sherrington-Kirkpatrick Hamiltonian, and so our lower bound implies a degree-n Omega(e) SoS lower bound for the certification version of the Sherrington-Kirkpatrick problem. The full version of the paper is available at http: //arxiv.org/abs/2009.01874.