Modified log-sobolev inequalities, mixing and hypercontractivity

Authors:
Sergey Bobkov;Prasad Tetali
Affiliations:
University of Minnesota, Minneapolis, MN;Georgia Tech, Atlanta, GA
Venue:
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Year:
2003

Citing 3
Cited 1

Time to reach stationarity in the Bernoulli-Laplace diffusion model

SIAM Journal on Mathematical Analysis
Elements of information theory

Elements of information theory
Concentration of Measure for Products of Markov Kernels and Graph Products via Functional Inequalities

Combinatorics, Probability and Computing

Robust mixing

APPROX'06/RANDOM'06 Proceedings of the 9th international conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th international conference on Randomization and Computation

Quantified Score

Hi-index	0.00

Visualization

Abstract

Motivated by (the rate of information loss or) the rate at which the entropy of an ergodic Markov chain relative to its stationary distribution decays to zero, we study modified versions of the standard logarithmic Sobolev inequality in the discrete setting of finite Markov chains and graphs. These inequalities turn out to be weaker than the standard log-Sobolev inequality, but stronger than the Poincare' (spectral gap) inequality. We also derive a hypercontractivity formulation equivalent to our main modified log-Sobolev inequality which might be of independent interest. Finally we show that, in contrast with the spectral gap, for bounded degree expander graphs various log-Sobolev-type constants go to zero with the size of the graph.