# Non-linear Log-Sobolev inequalities for the Potts semigroup and applications to reconstruction problems

@article{Gu2020NonlinearLI, title={Non-linear Log-Sobolev inequalities for the Potts semigroup and applications to reconstruction problems}, author={Yuzhou Gu and Yury Polyanskiy}, journal={ArXiv}, year={2020}, volume={abs/2005.05444} }

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…

## One Citation

### Sparse random hypergraphs: Non-backtracking spectra and community detection

- Computer Science, Mathematics
- 2022

To the best of the knowledge, this is the first provable and efficient spectral algorithm that achieves the conjectured threshold for HSBMs with r blocks generated according to a general symmetric probability tensor.

## References

SHOWING 1-10 OF 45 REFERENCES

### A symmetric entropy bound on the non-reconstruction regime of Markov chains on Galton-Watson trees

- Mathematics
- 2009

We give a criterion for the non-reconstructability of tree-indexed $q$-state Markov chains obtained by broadcasting a signal from the root with a given transition matrix $M$. Non-reconstruction holds…

### Improved log-Sobolev inequalities, hypercontractivity and uncertainty principle on the hypercube

- Mathematics, Computer ScienceJournal of Functional Analysis
- 2019

### On Reverse Hypercontractivity

- MathematicsGeometric and Functional Analysis
- 2013

We study the notion of reverse hypercontractivity. We show that reverse hypercontractive inequalities are implied by standard hypercontractive inequalities as well as by the modified log-Sobolev…

### LOGARITHMIC SOBOLEV INEQUALITIES FOR FINITE MARKOV CHAINS

- Mathematics
- 1996

This is an expository paper on the use of logarithmic Sobolev inequalities for bounding rates of convergence of Markov chains on finite state spaces to their stationary distributions. Logarithmic…

### Logarithmic Sobolev Inequalities

- Mathematics
- 2014

After Poincare inequalities, logarithmic Sobolev inequalities are amongst the most studied functional inequalities for semigroups. They contain much more information than Poincare inequalities, and…

### Modified Logarithmic Sobolev Inequalities in Discrete Settings

- Mathematics
- 2006

Motivated by the rate at which the entropy of an ergodic Markov chain relative to its stationary distribution decays to zero, we study modified versions of logarithmic Sobolev inequalities in the…

### Reconstruction and estimation in the planted partition model

- Mathematics, Computer Science
- 2015

This work establishes a rigorous connection between the clustering problem, spin-glass models on the Bethe lattice and the so called reconstruction problem and provides a simple and efficient algorithm for estimating a and b when clustering is possible.

### Reconstruction of Random Colourings

- Mathematics, Computer Science
- 2008

This work considers the reconstruction problem for random k-colourings on the Δ-ary tree for large k and shows non-reconstruction when$$Delta \geq k[\log k + \log \log k - o(1) + 1+o(1)]}$$, which is very close to the best known bound establishing reconstruction.

### A theorem on the entropy of certain binary sequences and applications-II

- Mathematics, Computer ScienceIEEE Trans. Inf. Theory
- 1973

A theorem concerning the entropy of a certain sequence of binary random variables is established and this result is applied to the solution of three problems in multi-user communication.