arXiv:2609.11836 · v2

Free-Probabilistic State Evolution and Random Matrix Discrepancy

首次提交 2026-09-11 01:28

版本更新 2026-11-04 01:10

雷达观测 2026-11-04 23:39

摘要

Let $A_1,\ldots,A_n$ be independent $d \times d$ real symmetric Gaussian random matrices, and consider the linear operator $A(x) = n^{-1/2}\sum_{i=1}^n x_iA_i$, $x\in \mathbb R^n$. We construct an iterative algorithm in the Approximate Message Passing family which iterates over $A$ and its adjoint $A^*$, and establish a state evolution result which characterizes its behavior in the limit $d\to \infty, 2n/d^2 \to α$ in terms of a correlated Gaussian-semicircular process in a free probability space, in the sense of strong convergence of operators. We then apply this iteration to the random matrix discrepancy problem which asks for a binary vector $x \in \{-1,+1\}^n$ such that $A(x)$ has a small operator norm. Our algorithm achieves an operator norm $2σ(α)$, for a standard deviation $σ(α)<1$ for all $α>0$, and $σ(α) = O(1/\sqrt{α\log α})$ for large $α$. This resolves the algorithmic question of Kunisky-Zhang (2023) and Maillard (2025).

作者备注：Revised version contains stronger results on random matrix discrepancy, further minor edits, and an updated bibliography