Sum-of-Squares Lower Bounds for Sherrington-Kirkpatrick via Planted Affine Planes
Mrinalkanti Ghosh, Fernando Granha Jeronimo, Chris Jones, Aaron Henry Potechin, Goutham Rajendran · 2020
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 d1, ..., dmin Rn, can we prove that there is no vector v ∈ IRnsuch that for all u ∈ [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 ≤ n3/2-ε, with high probability, degree- nΩ(ε)SoS fails to refute the existence of such a vector v. When the vectors d1, ..., dmare chosen from the multivariate normal distribution, the PAP problem is equivalent to the problem of proving that a random n-dimensional subspace of Rmdoes 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- nΩ(ε)SoS lower bound for the certification version of the Sherrington-Kirkpatrick problem.