凸序中的向量平衡
Vector balancing in convex order
浏览论文内容
中文总结 AI 辅助
本文在凸序框架下构造Komlós符号律，解决Reis--Rothvoss、Nutz--Wang--Zhang等多项猜想，给出体积比、私有发布等问题的普适界与准则。
中文摘要 AI 辅助
我们构造了偏差低于 $6.84$ 的 Komlós 符号律、独立高斯参考块和指数多个平衡符号。一个律保持硬约束和精确条件均值，而其参考控制每个联合凸成本。我们解决了 Reis--Rothvoss Schatten 猜想。对于 $n$ 个对称 $n\times n$ 矩阵，一个规定均值的符号律以 $n$ 的尖锐幂和普适常数界定所有中心 Schatten 偏差。一个测度为 $e^{-O(n)}$ 的高斯事件在平方函数半径处控制每个酉不变范数。我们证明了 Reis 的体积比猜想：$a\times b$ 迹类单位球的每个 $r$ 维商具有体积比 $O(\sqrt{1+\min(a,b)/r})$。我们解决了 Nutz--Wang--Zhang 的方向和超鞅猜想以及高维 Bass 流猜想。对于平方可积律，全凸参考支持是 Tschiderer 的 $q$-Bass 问题中普适有限势准则。每个驱动，甚至原子的，都承认一个扩展势和移位，用于每个具有全终端仿射跨度的不可约对的每个最大协方差鞅优化器。对于 Talagrand 问题，紧平衡高斯半质量集的五个副本包含质量至少 $3/4$ 的紧凸输出，在 Wiener 空间中也如此。Cheeger 界在混合加权乘积的次指数 Dirichlet 谱范围内渐近精确。对于标准 Borel 源，Blackwell 最大完美私有全数据发布存在当且仅当在丢弃条件点质量后，其余部分重合或共享一个两点支持。在条件无原子性下，存在等价于独立性。协方差匹配的解析一致对数凹源可以具有具有无限信息的最大的仅有用数据私有发布，而最大的全数据发布不存在。
英文摘要
We construct Komlós signing laws with discrepancy below $6.84$, independent Gaussian reference blocks and exponentially many balanced signings. One law preserves hard constraints and exact conditional means while its reference controls every joint convex cost. We resolve both Reis--Rothvoss Schatten conjectures. For $n$ symmetric $n\times n$ matrices, one prescribed-mean signing law bounds all centered Schatten discrepancies with sharp powers of $n$ and a universal constant. One Gaussian event of measure $e^{-O(n)}$ controls every unitarily invariant norm at the square-function radius. We prove Reis's volume-ratio conjecture: every $r$-dimensional quotient of the $a\times b$ trace-class unit ball has volume ratio $O(\sqrt{1+\min(a,b)/r})$. We resolve Nutz--Wang--Zhang's directional and supermartingale conjectures and the higher-dimensional Bass-flow conjecture. For square-integrable laws, full convex reference support is the universal finite-potential criterion in Tschiderer's $q$-Bass problem. Every driver, even atomic, admits one extended potential and shift for all maximal-covariance martingale optimizers of each irreducible pair with full terminal affine span. For Talagrand's problem, five copies of a compact balanced Gaussian half-mass set contain compact convex output of mass at least $3/4$, also in Wiener space. Cheeger's bound is asymptotically exact throughout subexponential Dirichlet spectral ranges of mixed weighted products. For standard Borel sources, a Blackwell-greatest perfectly private full-data release exists exactly when, after discarding conditional point masses, the rest coincide or share a two-point support. Under conditional atomlessness, existence is equivalent to independence. Covariance-matched analytic uniformly log-concave sources can have a greatest useful-data-only private release with infinite information, while no greatest full-data release exists.
