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
$d_{1}, \ldots, d_{m}$
in
$\mathrm{I}\!\mathrm{R}^{n}$
, can we prove that there is no vector
$v \in \mathrm{I}\!\mathrm{R}^{n}$
such that for all
$u\in[m],\ \langle v, d_{u}\rangle^{2}= 1$
? 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
$m\leq n^{3/ 2-\varepsilon}$
, with high probability, degree-
$n^{\Omega(\varepsilon)}$
SoS fails to refute the existence of such a vector
$v$
. When the vectors
$d_{1}, \ldots, d_{m}$
are chosen from the multivariate normal distribution, the PAP problem is equivalent to the problem of proving that a random
$n$
-dimensional subspace of
$\mathrm{I}\!\mathrm{R}^{m}$
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-
$n^{\Omega(\varepsilon)}$
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.