Derandomizing Matrix Concentration Inequalities from Free Probability

TL;DR

This paper develops polynomial time deterministic algorithms based on free probability to construct outcomes satisfying sharp matrix concentration inequalities, enabling solutions to problems like the matrix Spencer problem and near-Ramanujan graphs.

Contribution

It introduces the first polynomial time deterministic algorithms leveraging free probability for matrix concentration inequalities and related combinatorial problems.

Findings

Algorithms match the guarantees of recent matrix concentration inequalities

Deterministic construction of near-Ramanujan graphs in polynomial time

Application to the matrix Spencer problem with polynomial time solutions

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.