We develop new tools in the theory of nonlinear random matrices and apply them to study the performance of the Sum of Squares (SoS) hierarchy on average-case problems. The SoS hierarchy is a powerful optimization technique that has achieved tremendous success for various problems in combinatorial optimization, robust statistics and machine learning. It's a family of convex relaxations that lets us smoothly trade off running time for approximation guarantees. In recent works, it's been shown to be extremely useful for recovering structure in high dimensional noisy data. It also remains our best approach towards refuting the notorious Unique Games Conjecture. In this work, we analyze the performance of the SoS hierarchy on fundamental problems stemming from statistics, theoretical computer science and statistical physics. In particular, we show subexponential-time SoS lower bounds for the problems of the Sherrington-Kirkpatrick Hamiltonian, Planted Slightly Denser Subgraph, Tensor Principal Components Analysis and Sparse Principal Components Analysis. These SoS lower bounds involve analyzing large random matrices, wherein lie our main contributions. These results offer strong evidence for the truth of and insight into the low-degree likelihood ratio hypothesis, an important conjecture that predicts the power of bounded-time algorithms for hypothesis testing. We also develop general-purpose tools for analyzing the behavior of random matrices which are functions of independent random variables. Towards this, we build on and generalize the matrix variant of the Efron-Stein inequalities. In particular, our general theorem on matrix concentration recovers various results that have appeared in the literature. We expect these random matrix theory ideas to have other significant applications.

翻译：我们发展了非线性随机矩阵理论的新工具，并将其应用于研究平均情况问题中平方和（SoS）层级的性能。SoS层级是一种强大的优化技术，在组合优化、鲁棒统计和机器学习等多个问题上取得了巨大成功。它是一个凸松弛系列，能够让我们在运行时间与近似保证之间灵活权衡。近年研究表明，该技术在恢复高维含噪数据结构方面极为有效，且至今仍是反驳著名的唯一博弈猜想的最佳途径。在本工作中，我们分析了SoS层级在统计学、理论计算机科学和统计物理学基础问题上的性能表现。具体而言，我们针对Sherrington-Kirkpatrick哈密顿量、种植略密集子图、张量主成分分析和稀疏主成分分析等问题，证明了次指数时间SoS下界。这些SoS下界的推导涉及对大随机矩阵的分析，这也是本文的主要贡献。上述结果为低度似然比假设——一个预测有界时间算法在假设检验中能力的重要猜想——提供了强有力的证据支撑与深刻见解。我们还发展了分析作为独立随机变量函数的随机矩阵行为的通用工具。为此，我们基于并推广了Efron-Stein不等式的矩阵变体。特别地，关于矩阵集中性的通用定理复现了文献中的多种结果。我们预期这些随机矩阵理论的思想将具有其他重要应用。