正则单纯形对高斯最大值的随机支配
Stochastic Domination of Gaussian Maxima by the Regular Simplex
- University of Illinois Chicago（伊利诺伊大学芝加哥分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明正则单纯形在随机支配意义下最小化高斯最大值概率，并应用于等能量信号检测，证明其唯一最大化正确识别概率。
AI中文摘要：
设 $n\ge2$，且令 $X=(X_1,\ldots,X_n)$ 为中心高斯向量，满足对每个 $i$ 有 $\mathrm{Var}(X_i)=1$。令 $Z_1,\ldots,Z_n$ 为独立标准高斯变量，并设 $\overline{Z}=(Z_1+\cdots+Z_n)/n$。我们证明对每个 $t\in\mathbb{R}$ 有 $\mathbb{P}\{\max_i X_i\le t\}\ge\mathbb{P}\{\sqrt{n/(n-1)}\\,\max_i(Z_i-\overline{Z})\le t\}$，且对每个固定的 $t>0$，等号仅在当所有 $i\ne j$ 时 $\mathrm{Cov}(X_i,X_j)=-1/(n-1)$ 才成立。右侧是正则单纯形向量的最大值的分布函数。等价地，在所有包含给定中心球的单纯形中，外切于该球的正则单纯形具有最小的标准高斯测度，正如 Balitskiy、Karasev 和 Tsigler 所猜想的那样。在我们之前的论文中，我们在两个最大值都经过方差为 $1/(n-1)$ 的独立高斯噪声平滑后证明了这一比较，这足以证明弱单纯形猜想；在这里我们去掉了平滑，这正是单一阈值下概率所需要的。作为一个应用，我们考虑在高斯噪声中 $n$ 个等能量且等可能的信号，其中发射器也可能不发送任何内容。在每个正的虚警水平下，并且对于任何非负随机幅度（不集中于零）的公共分布，只要信号维度至少为 $n-1$，正则单纯形就唯一地最大化正确识别的平均概率。Lean 形式化可在 https://github.com/abhmul/full-simplex-conjecture-lean 获取。
英文摘要：
Let $n\ge2$, and let $X=(X_1,\ldots,X_n)$ be a centered Gaussian vector with $\mathrm{Var}(X_i)=1$ for every $i$. Let $Z_1,\ldots,Z_n$ be independent standard Gaussians, and put $\overline{Z}=(Z_1+\cdots+Z_n)/n$. We prove $\mathbb{P}\{\max_i X_i\le t\}\ge\mathbb{P}\{\sqrt{n/(n-1)}\,\max_i(Z_i-\overline{Z})\le t\}$ for every $t\in\mathbb{R}$, and for each fixed $t>0$ equality holds only when $\mathrm{Cov}(X_i,X_j)=-1/(n-1)$ for all $i\ne j$. The right side is the distribution function of the maximum of the regular simplex vector. Equivalently, among all simplices containing a given centered ball, the regular simplex circumscribed about the ball has the least standard Gaussian measure, as conjectured by Balitskiy, Karasev, and Tsigler. In our preceding paper we proved this comparison after both maxima are smoothed by independent Gaussian noise of variance $1/(n-1)$, which suffices for the Weak Simplex Conjecture; here we remove the smoothing, which is what probabilities at a single threshold require. As an application we consider $n$ equally likely signals of equal energy in Gaussian noise, where the transmitter may also send nothing. At every positive false-alarm level, and for every law of a common nonnegative random amplitude not concentrated at zero, the regular simplex uniquely maximizes the average probability of correct identification whenever the signal dimension is at least $n-1$. A Lean formalization is available at https://github.com/abhmul/full-simplex-conjecture-lean.
