Matrix concentration inequalities and their recently discovered sharp counterparts provide powerful tools to bound the spectrum of random matrices whose entries are linear functions of independent random variables. However, in many applications in theoretical computer science and in other areas one encounters more general random matrix models, called matrix chaoses, whose entries are polynomials of independent random variables. Such models have often been studied on a case-by-case basis using ad-hoc methods that can yield suboptimal dimensional factors. In this paper we provide general matrix concentration inequalities for matrix chaoses, which enable the treatment of such models in a systematic manner. These inequalities are expressed in terms of flattenings of the coefficients of the matrix chaos. We further identify a special family of matrix chaoses of combinatorial type for which the flattening parameters can be computed mechanically by a simple rule. This allows us to provide a unified treatment of and improved bounds for matrix chaoses that arise in a variety of applications, including graph matrices, Khatri-Rao matrices, and matrices that arise in average case analysis of the sum-of-squares hierarchy.

翻译：矩阵集中不等式及其新近发现的尖锐对应物为界定其元素是独立随机变量线性函数的随机矩阵谱提供了有力工具。然而，在理论计算机科学及其他领域的诸多应用中，人们会遇到更一般的随机矩阵模型，即矩阵混沌，其元素是独立随机变量的多项式。此类模型常通过特设方法进行个案研究，可能导致次优的维度因子。本文为矩阵混沌提供了一般性的矩阵集中不等式，使得能够以系统化方式处理此类模型。这些不等式通过矩阵混沌系数的扁平化参数表示。我们进一步识别了一类特殊的组合型矩阵混沌族，其扁平化参数可通过简单规则机械计算。这使我们能够对多种应用中出现的矩阵混沌（包括图矩阵、Khatri-Rao矩阵，以及平方和层次平均情况分析中出现的矩阵）进行统一处理并获得改进的界。