Problem
Research questions and friction points this paper is trying to address.
Random Matrix Discrepancy
Gaussian Random Matrices
Approximate Message Passing
State Evolution
Operator Norm
Free-Probabilistic State Evolution and Random Matrix Discrepancy
🤖 AI Summary
研究通过构建迭代算法解决随机矩阵不一致问题，利用自由概率空间中的状态演化结果，实现对特定条件下二元向量的高效搜索。
📝 Abstract
Let $A_1,\ldots,A_n$ be independent $d \times d$ real symmetric Gaussian random matrices, and consider the linear operator $A(x) = n^{-1/2}\sum_{i=1}^n x_i A_i$, $x\in \mathbb{R}^n$. 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 $d\rightarrow \infty, 2n/d^2 \rightarrow α$ 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 an explicit expression of the standard deviation $σ(α)<1$ for all $0<α<α_* \simeq 5.74$. This resolves the algorithmic question of Kunisky-Zhang (2023) and Maillard (2025) in this interval.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Approximate Message Passing
Free Probability Space
Random Matrix Discrepancy
State Evolution
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