发表机构
University of Warsaw（华沙大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明二阶高斯混沌的算子范数Sudakov下界，通过自适应高斯实验和凸分离方法，给出分离矩阵族期望上确界的下界估计。
AI 中文摘要
我们证明，一个按算子范数分离的矩阵族满足 $\mathbb{E}\sup_{A\in T} G^{T}AG' \geq ca\log |T|$，其中 $G,G'$ 是独立的标准高斯向量，$a$ 是分离度。主要的信息估计涉及任意分离的余等距算子：条件熵由一个依赖于源的算子能量乘以 $\log|T|$ 所界定，再加上公共行维数的二次项。一个自适应高斯实验通过将实际信息增量计入一个加权后验熵势来证明该估计。凸分离和高斯覆盖估计随后产生一个有界半径的结果。为达到一般情形，我们首先选择一个保持Sudakov比率的算子尺度，应用已知的Hilbert-Schmidt下界，并在保留的熵下重新计算一个公共高斯块压缩。这一顺序保持了余等距论证所需的归一化。
英文摘要
We prove that an operator-norm separated family of matrices satisfies $\mathbb{E}\sup_{A\in T} G^{T}AG' \geq ca\log |T|$, where G,G' are independent standard Gaussian vectors and a is the separation. The main information estimate concerns arbitrary separated coisometries: conditional entropy is bounded by a source-dependent operator energy times $\log|T|$, up to an additive quadratic term in the common row dimension. An adaptive Gaussian experiment proves this estimate by charging actual information increments to one weighted posterior-entropy potential. Convex separation and a Gaussian covering estimate then yield a bounded-radius result. To reach the general case, we first choose an operator scale preserving the Sudakov ratio, apply the known Hilbert-Schmidt minoration, and recompute a common Gaussian block compression at the retained entropy. This ordering preserves the normalization needed by the coisometry argument.
CommentsAI (GPT-6 Astra) was used in this research
