# Modified log-Sobolev inequalities for strongly log-concave distributions

## TL;DR

This paper establishes a lower bound on the modified log-Sobolev constant for certain strongly log-concave distributions, leading to improved mixing time and concentration bounds for related Markov chains.

## Contribution

It introduces a new lower bound on the modified log-Sobolev constant for r-homogeneous strongly log-concave distributions, with applications to mixing times and concentration inequalities.

## Key findings

- Lower bound of 1/r on the modified log-Sobolev constant.
- Sharp mixing time bounds for the bases-exchange walk for matroids.
- Concentration bounds for Lipschitz functions over these distributions.

## Abstract

We show that the modified log-Sobolev constant for a natural Markov chain which converges to an $r$-homogeneous strongly log-concave distribution is at least $1/r$. Applications include a sharp mixing time bound for the bases-exchange walk for matroids, and a concentration bound for Lipschitz functions over these distributions.