Let $P_Z$ be a given distribution on $\mathbb{R}^n$. For any $y\in\mathbb{R}^n$, we may interpret $\rho(y):=\ln\mathbb{E}[e^{\left<y,Z\right>}]$ as a soft-max of $\left<y,Z\right>$. We explore lower bounds on $\mathbb{E}[\rho(Y)]$ in terms of the minimum mutual information $I(Z,\bar{Z})$ over $P_{Z\bar{Z}}$ which is a coupling of $P_Z$ and itself such that $Z-\bar{Z}$ is bounded in a certain sense. This may be viewed as a soft version of Sudakov's minoration, which lower bounds the expected supremum of a stochastic process in terms of the packing number. Our method is based on convex geometry (thrifty approximation of convex bodies), and works for general non-Gaussian $Y$. When $Y$ is Gaussian and $\bar{Z}$ converges to $Z$, this recovers a recent inequality of Bai-Wu-Ozgur on information-constrained optimal transport, previously established using Gaussian-specific techniques. We also use soft-minoration to obtain asymptotically (in tensor order) tight bounds on the free energy in the Sherrington-Kirkpatrick model with spins uniformly distributed on a type class, implying asymptotically tight bounds for the type~II error exponent in spiked tensor detection.

翻译：设$P_Z$为$\mathbb{R}^n$上的给定分布。对于任意$y\in\mathbb{R}^n$，可将$\rho(y):=\ln\mathbb{E}[e^{\left<y,Z\right>}]$解释为$\left<y,Z\right>$的软最大值（soft-max）。我们探究$\mathbb{E}[\rho(Y)]$的下界，该下界基于$P_{Z\bar{Z}}$（$P_Z$与其自身的耦合）上的最小互信息$I(Z,\bar{Z})$，要求$Z-\bar{Z}$在某种意义上具有有界性。这可视作苏达科夫极小化（Sudakov's minoration）的软版本，后者以填充数（packing number）下界随机过程期望上确界。我们的方法基于凸几何（凸体的节俭近似），适用于一般非高斯的$Y$。当$Y$为高斯分布且$\bar{Z}$收敛于$Z$时，该方法可重构Bai-Wu-Ozgur近期关于信息约束最优输运的不等式，该不等式先前仅通过高斯特定技术建立。我们还利用软极小化获得谢林顿-柯克帕特里克（Sherrington-Kirkpatrick）模型中自旋均匀分布于类型类上的自由能的渐近（按张量阶）紧界，进而推导出尖峰张量检测中类型II错误指数的渐近紧界。