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 \emph{Planted Affine Planes} (PAP) problem: Given $m$ random vectors $d_1,\ldots,d_m$ in $\mathbb{R}^n$, can we prove that there is no vector $v \in \mathbb{R}^n$ 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 $m \leq n^{3/2-\epsilon}$, with high probability, degree-$n^{\Omega(\epsilon)}$ 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 $\mathbb{R}^m$ 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-$n^{\Omega(\epsilon)}$ SoS lower bound for the certification version of the Sherrington-Kirkpatrick problem.

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谢林顿-柯克帕特里克通过植入仿射平面的平方和下界

平方和 (SoS) 层次结构是一种半定编程元算法，它为许多优化问题（例如 Max-$k$-CSP 和 Tensor PCA）捕获了最先进的多项式时间保证。另一方面，SoS 下限提供了硬度的证据，这与 NP 硬度可能不可用的平均情况问题特别相关。在本文中，我们考虑以下平均情况问题，我们称之为 \emph{Planted Affine Planes} (PAP) 问题： 给定 $m$ 随机向量 $d_1,\ldots,d_m$ in $\mathbb{R}^ n$，我们能否证明没有向量 $v \in \mathbb{R}^n$ 使得对于所有 $u \in [m]$, $\langle v, d_u\rangle^2 = 1$？换句话说，我们能否证明 $m$ 个随机向量并非都包含在距原点相等距离的两个平行超平面中？我们证明，对于 $m \leq n^{3/2-\epsilon}$，很有可能，degree-$n^{\Omega(\epsilon)}$ SoS 无法反驳这样一个向量 $v 的存在$. 当向量$d_1,\ldots,d_m$从多元正态分布中选取时，PAP问题等价于证明$\mathbb{R}^m$的随机$n$维子空间不包含一个布尔向量。正如 Mohanty--Raghavendra--Xu [STOC 2020] 所示，这个问题的下限意味着证明 Sherrington-Kirkpatrick Hamiltonian 的能量上限问题的下限，因此我们的下限意味着度数-$ n^{\Omega(\epsilon)}$ 谢林顿-柯克帕特里克问题认证版本的 SoS 下界。d_m$是从多元正态分布中选取的，PAP问题等价于证明$\mathbb{R}^m$的随机$n$维子空间不包含布尔向量的问题。正如 Mohanty--Raghavendra--Xu [STOC 2020] 所示，这个问题的下限意味着证明 Sherrington-Kirkpatrick Hamiltonian 的能量上限问题的下限，因此我们的下限意味着度数-$ n^{\Omega(\epsilon)}$ 谢林顿-柯克帕特里克问题认证版本的 SoS 下界。d_m$是从多元正态分布中选取的，PAP问题等价于证明$\mathbb{R}^m$的随机$n$维子空间不包含布尔向量的问题。正如 Mohanty--Raghavendra--Xu [STOC 2020] 所示，这个问题的下限意味着证明 Sherrington-Kirkpatrick Hamiltonian 的能量上限问题的下限，因此我们的下限意味着度数-$ n^{\Omega(\epsilon)}$ 谢林顿-柯克帕特里克问题认证版本的 SoS 下界。