38 research outputs found
Elementary bounds on Poincare and log-Sobolev constants for decomposable Markov chains
We consider finite-state Markov chains that can be naturally decomposed into
smaller ``projection'' and ``restriction'' chains. Possibly this decomposition
will be inductive, in that the restriction chains will be smaller copies of the
initial chain. We provide expressions for Poincare (resp. log-Sobolev)
constants of the initial Markov chain in terms of Poincare (resp. log-Sobolev)
constants of the projection and restriction chains, together with further a
parameter. In the case of the Poincare constant, our bound is always at least
as good as existing ones and, depending on the value of the extra parameter,
may be much better. There appears to be no previously published decomposition
result for the log-Sobolev constant. Our proofs are elementary and
self-contained.Comment: Published at http://dx.doi.org/10.1214/105051604000000639 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Modified log-Sobolev inequalities for strongly log-concave distributions
We show that the modified log-Sobolev constant for a natural Markov chain
which converges to an -homogeneous strongly log-concave distribution is at
least . Applications include a sharp mixing time bound for the
bases-exchange walk for matroids, and a concentration bound for Lipschitz
functions over these distributions.Comment: accepted to Annals of Probability. Simplified proof