亚高斯坐标对称随机张量的强凸集中性
Sharp Convex Concentration for Symmetric Random Tensors with Subgaussian Coordinates
AI总结：
该论文研究亚高斯坐标对称随机张量的凸函数集中性，推导了集中不等式的极小极大最优速率，通过耦合与二阶估计完成证明，为随机张量分析提供了严格的概率界。
AI中文摘要：
设$X=(X_1,\boldsymbol{\text{...}},X_n)$为独立坐标，满足均值为0、方差为1且$\boldsymbol{\text{||}}X_i\boldsymbol{\text{||}}_{\boldsymbol{\text{ψ}}_2}\boldsymbol{\text{≤}}K$，令$H_d=(\boldsymbol{\text{R}}^n)^{\boldsymbol{\text{⊗}}_2 d}$。设$L>0$，$f:H_d\to\boldsymbol{\text{R}}$为凸函数且是$L$-Lipschitz的。我们证明，当$0\boldsymbol{\text{≤}}t\boldsymbol{\text{≤}}c_KLn^{d/2}$时，$\boldsymbol{\text{P}}\boldsymbol{\text{\big{\text{|}}}f(X^{\boldsymbol{\text{⊗}}d})-\boldsymbol{\text{E}}f(X^{\boldsymbol{\text{⊗}}d})\boldsymbol{\text{|}}>t\boldsymbol{\text{\big{\text{|}}}}\boldsymbol{\text{≤}}C\boldsymbol{\text{exp}}\boldsymbol{\text{\big[-}}c_K\boldsymbol{\text{I}}_{n,d}\boldsymbol{\text{\big(}}\frac{t}{L n^{(d-1)/2}}\boldsymbol{\text{\big)}}\boldsymbol{\text{\big]}}$，其中$\boldsymbol{\text{I}}_{n,d}(s)=\boldsymbol{\text{min}}\boldsymbol{\text{\big{\text{}}}}\frac{s^2}{d^2},\frac{s^2}{d\boldsymbol{\text{log}}(e+nd/s^2)}\boldsymbol{\text{\big{\text{}}}}$（$s>0$，$\boldsymbol{\text{I}}_{n,d}(0)=0$）。第一种速率由$\boldsymbol{\text{||}}X\boldsymbol{\text{||}}$的变化决定，第二种由$X$范数近固定时$X$的变化产生。证明构造一个耦合同时控制坐标条件位移和均方欧氏距离，并结合$x\boldsymbol{\text{↦}}x^{\boldsymbol{\text{⊗}}d}$的二阶估计。该速率在各尺度上达到极小极大最优，即使亚高斯范数被绝对常数界定也成立；对于有界坐标，第二种速率中的对数项消失。
英文摘要：
Let $X=(X_1,\ldots,X_n)$ have independent coordinates with mean zero, variance one, and $\|X_i\|_{ψ_2}\le K$, and let $H_d=(\mathbb R^n)^{\otimes_2 d}$. Let $L>0$ and let $f:H_d\to\mathbb R$ be convex and $L$-Lipschitz. We prove that, for $0\le t\le c_KLn^{d/2}$, \[ \textsf{P}\left\{ \left\lvert f(X^{\otimes d})-\textsf{E}f(X^{\otimes d})\right\rvert >t \right\} \le C\exp\left[-c_K\mathcal I_{n,d}\left( \frac{t}{L n^{(d-1)/2}} \right)\right], \] where \[ \mathcal I_{n,d}(s)= \min\left\{ \frac{s^2}{d^2}, \frac{s^2}{d\log(e+nd/s^2)} \right\},\qquad s>0, \qquad \mathcal I_{n,d}(0)=0. \] The first rate is forced by changes in $\|X\|$. The second comes from changes of $X$ when its norm is nearly fixed. The proof constructs one coupling that controls both the coordinatewise conditional displacement and the mean squared Euclidean distance, and combines these bounds with a second-order estimate for $x\mapsto x^{\otimes d}$. The rate is minimax sharp, scale by scale, even when the subgaussian norms are bounded by an absolute constant. For bounded coordinates the logarithm in the second rate disappears.
