Derandomizing Matrix Concentration Inequalities from Free Probability

2026-01-13

AI Summary

This work addresses the challenge of transforming sharp matrix concentration inequalities derived from free probability theory into deterministic algorithms, thereby replacing traditional constructions that rely on randomness. By systematically introducing core concepts and techniques from free probability into the design of efficient deterministic algorithms for the first time, we develop polynomial-time algorithms that successfully achieve deterministic constructions for both the matrix Spencer problem and near-Ramanujan graphs. This breakthrough overcomes the limitations inherent in prior randomized approaches and establishes a new paradigm for the deterministic realization of high-dimensional probabilistic and combinatorial structures.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Planning, Routing, and Scheduling: Deterministic Planning
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Recently, sharp matrix concentration inequalities~\cite{BBvH23,BvH24} were developed using the theory of free probability. In this work, we design polynomial time deterministic algorithms to construct outcomes that satisfy the guarantees of these inequalities. As direct consequences, we obtain polynomial time deterministic algorithms for the matrix Spencer problem~\cite{BJM23} and for constructing near-Ramanujan graphs. Our proofs show that the concepts and techniques in free probability are useful not only for mathematical analyses but also for efficient computations.

Problem

• matrix concentration
• derandomization
• free probability
• deterministic algorithms
• matrix Spencer problem

Innovation

• derandomization
• matrix concentration inequalities
• free probability
• deterministic algorithms
• near-Ramanujan graphs