从自由概率到矩阵差异性的漫步 II：Weaver 问题与 Kadison-Singer 猜想
A Walk From Free Probability to Matrix Discrepancy II: Weaver's Problem and the Kadison-Singer Conjecture
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中文总结 AI 辅助
本文提出一个多项式时间确定性算法，通过超立方体漫步和自由概率势函数，解决 Weaver 问题并实现差异性至多 $35\sqrt\varepsilon$，为 Kadison-Singer 猜想提供构造性证明。
中文摘要 AI 辅助
\cite{mss2015} 以存在性方式证明了 Weaver 的差异性结果，从而解决了 Kadison-Singer 猜想。对于一般输入，高效地找到这样的符号仍然是一个开放性的算法问题。在实数算术模型中，我们给出一个确定性算法，其运行时间为多项式时间，差异性至多为 $35\sqrt\varepsilon$。该算法从超立方体的原点漫步至一个顶点，在坐标碰到面时将其固定。其势函数度量了由算子值自由半圆元素扰动的差异性矩阵的软谱边缘。当系数达到其端点时，扰动的协方差消失。受 Bandeira、Boedihardjo 和 van Handel 的自由插值方法启发，我们将 Lehner 的变分公式与 \cite{allenZhuLiaoOrecchia2015} 和 \cite{pesentivladu2026} 中使用的谱 Tsallis-$1/2$ 正则化相结合。由此产生的势函数具有有限维半定规划（SDP）形式，使得差异性和剩余协方差可以一起分析。我们通过正则化极小-极大问题的线性化 Karush-Kuhn-Tucker（KKT）系统来分析优化器的稳定性，其平稳性方程与矩阵 Dyson 方程相关。这给出了移动规则：要么一个坐标可以以较小的谱代价向其较近的端点移动，要么存在一个与当前系数向量正交的低曲率方向，允许进一步进展。选择该方向的较优符号可以控制差异性，同时增加与原点距离的平方。即将发表的工作将处理更高秩的 Kadison-Singer 问题和谱薄树。我们主要差异性定理的 Lean 形式化证明已经完成，并将很快发布。
英文摘要
\cite{mss2015} proved Weaver's discrepancy result existentially, resolving the Kadison--Singer conjecture . Finding such signs efficiently for general inputs remained an open algorithmic question. In the real-arithmetic model, we give a deterministic algorithm running in polynomial time with discrepancy at most $35\sqrt\varepsilon$. The algorithm walks from the origin of the hypercube to a vertex, fixing coordinates as they hit a face. Its potential measures a soft spectral edge of the discrepancy matrix perturbed by an operator-valued free semicircular element. The perturbation's covariance vanishes as the coefficients reach their endpoints. Inspired by the free interpolation approach of Bandeira, Boedihardjo, and van Handel \cite{bbvh2023}, we combine Lehner's variational formula \cite{lehner1999} with spectral Tsallis--$1/2$ regularization used in \cite{allenZhuLiaoOrecchia2015} and \cite{pesentivladu2026}. The resulting potential has a finite-dimensional SDP formulation, allowing the discrepancy and remaining covariance to be analyzed together. We analyze the optimizer's stability through the linearized Karush--Kuhn--Tucker (KKT) system of a regularized min--max problem, whose stationarity equations are related to the matrix Dyson equation \cite{erdos2019}. This gives the movement rule: either a coordinate can move toward its nearer endpoint at small spectral cost, or a low-curvature direction orthogonal to the current coefficient vector allows further progress. Choosing the better sign of this direction controls discrepancy while increasing the squared distance from the origin. Upcoming work \cite{kathuria2026higherRank} will address higher-rank Kadison-Singer and spectrally thin trees. Lean formalizations of our main discrepancy theorems have been completed and will be released shortly.
发表机构
- Google（谷歌）
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