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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 termsof the minimum mutual information $I(Z,\bar{Z})$ over $P_{Z\bar{Z}}$ which is acoupling of $P_Z$ and itself such that $Z-\bar{Z}$ is bounded in a certainsense. This may be viewed as a soft version of Sudakov's minoration, whichlower bounds the expected supremum of a stochastic process in terms of thepacking number. Our method is based on convex geometry (thrifty approximationof convex bodies), and works for general non-Gaussian $Y$. When $Y$ is Gaussianand $\bar{Z}$ converges to $Z$, this recovers a recent inequality ofBai-Wu-Ozgur on information-constrained optimal transport, previouslyestablished using Gaussian-specific techniques. We also use soft-minoration toobtain asymptotically (in tensor order) tight bounds on the free energy in theSherrington-Kirkpatrick model with spins uniformly distributed on a type class,implying asymptotically tight bounds for the type~II error exponent in spikedtensor detection.

From Soft-Minoration to Information-Constrained Optimal Transport and Spiked Tensor Models