发表机构
University of Illinois Urbana-Champaign; Yale University（伊利诺伊大学厄巴纳-香槟分校; 耶鲁大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究随机矩阵差异问题，提出稳定离线算法（recenter-and-round）和在线算法（Frobenius-greedy），确定其相变阈值分别为Θ(1/(κ²log(1/κ)))和Θ(1/κ²)，均高于可满足性尺度。
AI 中文摘要
我们研究平均情况下的矩阵差异问题：给定独立的归一化 $d\times d$ 高斯正交系综矩阵 $A_1,\dots,A_N$ 以及固定裕度 $\kappa>0$，寻找符号 $\sigma_1,\dots,\sigma_N\in\{-1,1\}$ 使得 $\sum_{i=1}^N \sigma_i A_i$ 的算子范数至多为 $\kappa\sqrt{N}$。关注比例区间 $N/d^2\to \tau\in(0,\infty)$（当 $d\to\infty$ 时）随后是小裕度极限 $\kappa\downarrow 0$，我们刻画了稳定离线算法和在线算法所需的密度。在离线设置中，我们构造了一个多项式时间的“重新居中并舍入”（recenter-and-round）算法，该算法具有噪声稳定性，并且当 $\tau=\Omega(\frac{1}{\kappa^2\log(1/\kappa)})$ 时成功，同时给出了所有稳定算法的匹配下界。在在线设置中，每个符号必须在观察到相应矩阵时不可撤销地选择，我们确定了“Frobenius-贪心”（Frobenius-greedy）算法的精确极限性能，证明当 $\tau>\tau_{\rm FG}(\kappa)\sim \frac{\pi}{4\kappa^2}$ 时成功，同时通过条件于已揭示前缀给出了所有在线算法的匹配下界。我们算法的核心在于旋转对称性，这使我们能够将 Frobenius 范数控制转化为算子范数保证。综合起来，我们的结果确定了随机矩阵差异的算法相变点：稳定离线算法为 $\Theta(\frac{1}{\kappa^2\log(1/\kappa)})$，在线算法为 $\Theta(\frac{1}{\kappa^2})$。这两个阈值都远高于可满足性尺度 $\Theta(\log(1/\kappa))$，如 Maillard~\cite{maillard2025} 所示。
英文摘要
We study the average-case matrix discrepancy problem: given independent normalized $d\times d$ Gaussian orthogonal ensemble matrices $A_1,\dots,A_N$ and a fixed margin $κ>0$, find signs $σ_1,\dots,σ_N\in\{-1,1\}$ such that the operator norm of $\sum_{i=1}^N σ_i A_i$ is at most $κ\sqrt{N}$. Focusing on the proportional regime $N/d^2\to τ\in(0,\infty)$ as $d\to\infty$ followed by the small-margin limit $κ\downarrow 0$, we characterize the density required by stable offline algorithms and by online algorithms. In the offline setting, we construct a polynomial-time \emph{recenter-and-round} algorithm that is noise-stable and succeeds whenever $τ=Ω(\frac{1}{κ^2\log(1/κ)})$, along with a matching lower bound for all stable algorithms. In the online setting where each sign must be chosen irrevocably upon observing the corresponding matrix, we determine the exact limiting performance of the \emph{Frobenius-greedy} algorithm, establishing that it succeeds when $τ>τ_{\rm FG}(κ)\sim \fracπ{4κ^2}$, as well as a matching lower bound for all online algorithms by conditioning on a revealed prefix. At the core of our algorithms lies rotational symmetry, which enables us to transfer Frobenius norm control into operator norm guarantees. Together, our results identify the algorithmic phase transition points for random matrix discrepancy: $Θ(\frac{1}{κ^2\log(1/κ)})$ for stable offline algorithms and $Θ(\frac{1}{κ^2})$ for online algorithms. Both thresholds lie far above the satisfiability scale $Θ(\log(1/κ))$, as shown by Maillard~\cite{maillard2025}.
