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 NP-hardness 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 R n , can we prove that there is no vector v ∈ R n such that for all u ∈ [m], v, d u 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-ε , with high probability, degree-n Ω(ε) 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 R m does not contain a boolean vector. As shown by Mohanty-Raghavendra-Xu [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 Ω(ε) SoS lower bound for the certification version of the Sherrington-Kirkpatrick problem.