Exponents for Concentration of Measure and Isoperimetry in Product Spaces
Abstract
In this paper, we provide variational formulas for the asymptotic exponents of the concentration function in the product probability space. The variational formulas for the exponents are expressed in terms of relative entropies (which are from information theory) and optimal transport cost functionals (which are from optimal transport theory). Moreover, in the concentration of measure regime, our variational formula is in fact a dimensionfree bound on the concentration function, which is valid for any finite dimensions. Our proofs in this paper are based on informationtheoretic and optimal transport techniques, and our results verify an intimate connection among information theory, optimal transport, and concentration of measure.
 Publication:

arXiv eprints
 Pub Date:
 May 2022
 arXiv:
 arXiv:2205.07596
 Bibcode:
 2022arXiv220507596Y
 Keywords:

 Mathematics  Probability;
 Computer Science  Information Theory;
 Mathematics  Functional Analysis;
 Mathematics  Metric Geometry
 EPrint:
 27 pages, preliminary version