Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Non-linear Log-Sobolev inequalities for the Potts semigroup and
  applications to reconstruction problems   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 Saloff-Coste computed the maximum of the ratio of the
relative entropy and the Dirichlet form obtaining the constant $\alpha_2$ 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, $p\ge1$). 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 Fourier-analytic 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 non-trivial
reconstruction probability the branching number of the tree should be at least
$$\frac{\log k}{\log k - \log(k-1)} = (1-o(1))k\log k.$$ 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 state-of-the-art on
reconstruction threshold for the stochastic block model with $k$ balanced
groups, for all $k\ge 3$. These improvements advocate information-theoretic
methods as a useful complement to the conventional techniques originating from
the statistical physics.

18
The categories of this paper are: econ
11
The categories of this paper are: hep-lat
29
The categories of this paper are: funct-an
22
The categories of this paper are: chao-dyn
13
The categories of this paper are: quant-ph
26
The categories of this paper are: bayes-an
27
The categories of this paper are: comp-gas
25
The categories of this paper are: patt-sol
9
The categories of this paper are: nlin
35
The categories of this paper are: mtrl-th
30
The categories of this paper are: q-alg
4
The categories of this paper are: cond-mat
19
The categories of this paper are: eess
37
The categories of this paper are: supr-con
21
The categories of this paper are: adap-org
14
The categories of this paper are: math-ph