Non-linear Log-Sobolev inequalities for the Potts semigroup and applications to reconstruction problems
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Consider a Markov process with state space [k], which jumps continuously to a new state chosen uniformly at random and regardless of the previous state. The collection of transition kernels (indexed by time t> 0) is the Potts semigroup. Diaconis and Saloff-Coste computed the maximum of the ratio of the relative entropy and the Dirichlet form obtaining the constant α_2 in the 2-log-Sobolev inequality (2-LSI). In this paper, we obtain the best possible non-linear inequality relating entropy and the Dirichlet form (i.e., p-NLSI, p>1). As an example, we show α_1 = 1+1+o(1)/log k. The more precise NLSIs have been shown by Polyanskiy and Samorodnitsky to imply various geometric and Fourier-analytic results. Beyond the Potts semigroup, we also analyze Potts channels – Markov transition matrices [k]× [k] constant on and off diagonal. (Potts semigroup corresponds to a (ferromagnetic) subset of matrices with positive second eigenvalue). By integrating the 1-NLSI we obtain the new strong data processing inequality (SDPI), which in turn allows us to improve results on reconstruction thresholds for Potts models on trees. A special case is the problem of reconstructing color of the root of a k-colored tree given knowledge of colors of all the leaves. We show that to have a non-trivial reconstruction probability the branching number of the tree should be at least log k/log k - log(k-1) = (1-o(1))klog k. This extends previous results (of Sly and Bhatnagar et al.) to general trees, and avoids the need for any specialized arguments. Similarly, we improve the state-of-the-art on reconstruction threshold for the stochastic block model with k balanced groups, for all k> 3. These improvements advocate information-theoretic methods as a useful complement to the conventional techniques originating from the statistical physics.
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