We introduce a new method for building higher-degree sum-of-squares lower bounds over the hypercube $\mathbf{x} \in \{\pm 1\}^N$ from a given degree 2 lower bound. Our method constructs pseudoexpectations that are positive semidefinite by design, lightening some of the technical challenges common to other approaches to SOS lower bounds, such as pseudocalibration. We give general "incoherence" conditions under which degree 2 pseudomoments can be extended to higher degrees. As an application, we extend previous lower bounds for the Sherrington-Kirkpatrick Hamiltonian from degree 4 to degree 6. (This is subsumed, however, in the stronger results of the parallel work of Ghosh et al.) This amounts to extending degree 2 pseudomoments given by a random low-rank projection matrix. As evidence in favor of our construction for higher degrees, we also show that random high-rank projection matrices (an easier case) can be extended to degree $\omega(1)$. We identify the main obstacle to achieving the same in the low-rank case, and conjecture that while our construction remains correct to leading order, it also requires a next-order adjustment. Our technical argument involves the interplay of two ideas of independent interest. First, our pseudomoment matrix factorizes in terms of certain multiharmonic polynomials. This observation guides our proof of positivity. Second, our pseudomoment values are described graphically by sums over forests, with coefficients given by the M\"{o}bius function of a partial ordering of those forests. This connection guides our proof that the pseudomoments satisfy the hypercube constraints. We trace the reason that our pseudomoments can satisfy both the hypercube and positivity constraints simultaneously to a combinatorial relationship between multiharmonic polynomials and this M\"{o}bius function.

中文翻译：

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超立方体上平方和伪矩的保正性扩展

我们引入了一种新方法，用于从给定的 2 次下界在超立方体 $\mathbf{x} \in \{\pm 1\}^N$ 上构建更高次的平方和下界。我们的方法构建了设计为半正定的伪期望，减轻了其他 SOS 下界方法常见的一些技术挑战，例如伪校准。我们给出了一般的“不相干”条件，在这种条件下，2 级伪矩可以扩展到更高的程度。作为一个应用，我们将 Sherrington-Kirkpatrick Hamiltonian 的先前下界从 4 阶扩展到 6 阶。（然而，这被包含在 Ghosh 等人的并行工作的更强结果中。）这相当于扩展了 2 阶伪矩由随机低秩投影矩阵给出。作为支持我们建设更高学位的证据，我们还表明随机高秩投影矩阵（更简单的情况）可以扩展到 $\omega(1)$ 度。我们确定了在低秩情况下实现相同目标的主要障碍，并推测虽然我们的构造对前序保持正确，但它也需要对下序进行调整。我们的技术论点涉及两个独立兴趣的想法的相互作用。首先，我们的伪矩矩阵根据某些多谐多项式进行分解。这一观察指导我们证明积极性。其次，我们的伪矩值由森林的总和以图形方式描述，系数由这些森林的偏序的 M\"{o}bius 函数给出。这种联系指导我们证明伪矩满足超立方体约束的证明。