从自由概率到矩阵差异性的漫步 I：矩阵Spencer猜想
A Walk From Free Probability to Matrix Discrepancy I: Matrix Spencer
- Google（谷歌）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出随机化算法，通过协方差控制随机游走与自由概率势函数，在多项式时间内证明矩阵Spencer猜想，并给出Weaver定理的算法证明。
AI中文摘要：
矩阵Spencer猜想询问：任意n个算子范数至多为1的实对称矩阵A_1,...,A_n \in \mathbb{R}^{m \times m}，是否存在符号选择x\in\{-1,1\}^n，使得带符号求和的算子范数至多为O(\sqrt{n \log(2m/n)})。我们给出一个随机化算法，在实数算术模型中用多项式运行时间建立此界。我们首先证明m\le n情况下的O(\sqrt n)界，解决方形情形，然后通过改变正则化器获得矩形界。与早期算法差异性方法（如lovettmeka2012、bansalLaddhaVempala2022、pesentivladu2026）类似，我们从超立方体原点运行协方差控制的随机游走，在面附近舍入坐标并保持固定。我们的势函数度量由算子值自由半圆元素扰动的演化差异性矩阵的软谱边缘。受bbvh2023的自由插值方法启发，我们将Lehner的自由边缘变分公式与谱Tsallis正则化（allenZhuLiaoOrecchia2015、pesentivladu2026）相结合。这将差异性和剩余协方差置于单一光滑优化问题中。势函数具有有限维半定规划形式。其优化器的稳定性由与矩阵Dyson方程相关的方程控制，使我们能够找到一个大子空间在其中移动同时控制差异性。方形情形使用Tsallis-1/2正则化器；矩形情形使用合适的广义Tsallis幂正则化器。我们的姊妹论文kathuria2026ks应用这些思想给出Weaver差异性定理的算法证明，该定理的存在性证明由MSS15解决了Kadison-Singer猜想。我们主要差异性定理的Lean形式化已完成，并将很快发布。
英文摘要：
The Matrix Spencer conjecture asks whether any $n$ real symmetric matrices A_1,...,A_n \in \mathbb{R}^{m \times m} of operator norm at most one admit a signing $x\in\{-1,1\}^n$ such that the operator norm of the signed sum is at most O(\sqrt{n \log(2m/n)}) We give a randomized algorithm establishing this bound with polynomial runtime in the real-arithmetic model. We first prove the $O(\sqrt n)$ bound for $m\le n$, resolving the square case, and then obtain the rectangular bound by changing the regularizer. As in earlier algorithmic discrepancy methods \cite{lovettmeka2012,bansalLaddhaVempala2022,pesentivladu2026}, we run a covariance-controlled random walk from the origin of the hypercube, rounding coordinates near its faces and keeping them fixed. Our potential measures a soft spectral edge of the evolving discrepancy matrix perturbed by an operator-valued free semicircular element. Inspired by the free interpolation approach of \cite{bbvh2023}, we combine Lehner's variational formula for the free edge \cite{lehner1999} with spectral Tsallis regularization \cite{allenZhuLiaoOrecchia2015,pesentivladu2026}. This puts the discrepancy and remaining covariance in a single smooth optimization problem. The potential has a finite-dimensional semidefinite formulation. Stability of its optimizer, governed by equations related to the matrix Dyson equation \cite{erdos2019}, lets us find a large subspace in which to move while controlling discrepancy. The square case uses the Tsallis--$1/2$ regularizer; the rectangular case uses a suitable generalized Tsallis power regularizer. Our companion paper \cite{kathuria2026ks} applies these ideas to give an algorithmic proof of Weaver's discrepancy theorem, whose existence proof by [MSS15] resolved the Kadison--Singer conjecture \cite{mss2015}.Lean formalizations of our main discrepancy theorems have been completed and will be released shortly.
