Nonlinear LogSobolev inequalities for the Potts semigroup and applications to reconstruction problems
Abstract
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 $t\ge 0$) is the Potts semigroup. Diaconis and SaloffCoste computed the maximum of the ratio of the relative entropy and the Dirichlet form obtaining the constant $\alpha_2$ in the $2$logSobolev inequality ($2$LSI). In this paper, we obtain the best possible nonlinear inequality relating entropy and the Dirichlet form (i.e., $p$NLSI, $p\ge1$). As an example, we show $\alpha_1 = 1+\frac{1+o(1)}{\log k}$. The more precise NLSIs have been shown by Polyanskiy and Samorodnitsky to imply various geometric and Fourieranalytic results. Beyond the Potts semigroup, we also analyze Potts channels  Markov transition matrices $[k]\times [k]$ 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 nontrivial reconstruction probability the branching number of the tree should be at least $$\frac{\log k}{\log k  \log(k1)} = (1o(1))k\log k.$$ 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 stateoftheart on reconstruction threshold for the stochastic block model with $k$ balanced groups, for all $k\ge 3$. These improvements advocate informationtheoretic methods as a useful complement to the conventional techniques originating from the statistical physics.
 Publication:

arXiv eprints
 Pub Date:
 May 2020
 arXiv:
 arXiv:2005.05444
 Bibcode:
 2020arXiv200505444G
 Keywords:

 Computer Science  Information Theory;
 Mathematics  Probability;
 Mathematics  Statistics Theory