Matrix Chaos Inequalities and Chaos of Combinatorial Type

2024-12-24

AI Summary

Spectral concentration of high-order random matrices—termed “matrix chaos”—remains a fundamental challenge in theoretical computer science, particularly in average-case analysis of the Sum-of-Squares (SoS) hierarchy. Existing techniques rely heavily on restrictive linear or low-degree assumptions, failing to handle general polynomial matrix ensembles. Method: We develop the first general spectral bound theory for arbitrary polynomial random matrices, bypassing traditional linearity constraints. Our approach introduces a unified matrix concentration inequality based on coefficient tensor flattenings, and identifies a combinatorially structured class of matrix chaos whose parameters are mechanically computable via tensor decomposition and noncommutative probability tools. Contribution/Results: The framework yields automated, order-optimal spectral bounds with improved dimension dependence for graph matrices, Khatri–Rao products, and SoS average-case analysis. It provides a systematic theoretical toolkit for efficient and precise spectral analysis of complex random combinatorial systems.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Computational social science
• Security and Privacy: Large-scale security measurements

Abstract

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.

Problem

• Complex Random Combinatorics
• Chaos Matrix
• Computational Science Problems

Innovation

• Chaos Matrix Complexity
• Graph Matrices
• Khatri-Rao Matrices