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Derandomizing Matrix Concentration Inequalities from Free Probability

Robert Wang, Lap Chi Lau, Hong Zhou

TL;DR

This work derandomizes sharp matrix concentration inequalities rooted in free probability, providing polynomial-time deterministic algorithms that replicate probabilistic spectral guarantees. By introducing a free-probability–guided framework (X_free) and a Brownian/interpolation scheme, the authors replace portions of the free model with finite deterministic updates, enabling constructive outcomes that satisfy matrix-norm and spectral bounds. Central contributions include deterministic partial coloring for matrix discrepancy and matrix Spencer problems, full-spectrum derandomization, and deterministic expander constructions, all under a unified framework that leverages intrinsic freeness and resolvent-based analysis. These results offer practical, scalable tools for spectral sparsification, planted-recovery models, and expander design, substantiating the computational utility of free probability in algorithmic contexts.

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.

Derandomizing Matrix Concentration Inequalities from Free Probability

TL;DR

This work derandomizes sharp matrix concentration inequalities rooted in free probability, providing polynomial-time deterministic algorithms that replicate probabilistic spectral guarantees. By introducing a free-probability–guided framework (X_free) and a Brownian/interpolation scheme, the authors replace portions of the free model with finite deterministic updates, enabling constructive outcomes that satisfy matrix-norm and spectral bounds. Central contributions include deterministic partial coloring for matrix discrepancy and matrix Spencer problems, full-spectrum derandomization, and deterministic expander constructions, all under a unified framework that leverages intrinsic freeness and resolvent-based analysis. These results offer practical, scalable tools for spectral sparsification, planted-recovery models, and expander design, substantiating the computational utility of free probability in algorithmic contexts.

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.
Paper Structure (79 sections, 84 theorems, 452 equations)

This paper contains 79 sections, 84 theorems, 452 equations.

Table of Contents

  1. Introduction
  2. Technical Review
  3. Our Results
  4. Norms of Eigenvalues
  5. Deterministic Matrix Discrepancy
  6. Full Spectrum
  7. General Model
  8. Deterministic Expander Constructions
  9. Proof Overview
  10. Derandomizing Random Gaussian Matrix Models by Brownian Walks
  11. Derandomizing General Random Matrix Models by Random Swap
  12. Free Probability and Random Matrices
  13. An Introduction to Free Probability
  14. Algebra and Probability Space
  15. Distributions and Convergence
  16. ...and 64 more sections

Key Result

Theorem 1.1

Let $A_1, \ldots, A_n$ be $d \times d$ Hermitian matrices. Let $\mathcal{H} \subseteq \mathbb R^n$ be a linear subspace of dimension $(1-{\varepsilon})n$. For any $p \geq 4$, there is a deterministic polynomial time algorithm to find a vector $x \in [-1,1]^n \cap \mathcal{H}$ with $|\{i \mid x(i) \i and $\sigma(X), \nu(X)$ are defined as in e:sigma, e:nu respectively.

Theorems & Definitions (150)

  • Theorem 1.1: Deterministic Partial Coloring, Simplified Version of \ref{['t:partial-coloring-full']}
  • Theorem 1.2: Deterministic Matrix Spencer
  • Theorem 1.3: Deterministic Full Spectrum, Simplified Version of \ref{['t:spectrum-full']}
  • Theorem 1.4: Deterministic Moment Universality, Simplified Version of \ref{['t:moment-uni']}
  • Theorem 1.5: Deterministic Norm Universality, Simplified Version of \ref{['t:norm-uni']}
  • Definition 2.1: Non-Commutative Probability Space
  • Example 2.2: Random Matrices
  • Definition 2.3: $C^*$-Probability Space
  • Theorem 2.4: Functional Calculus and Spectral Mapping Theorem
  • Definition 2.5: Analytical Distribution
  • ...and 140 more