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}.

翻译：暂无翻译