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