Exponents for Concentration of Measure and Isoperimetry in Product Spaces
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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 dimension-free bound on the concentration function, which is valid for any finite dimensions. Our proofs in this paper are based on information-theoretic and optimal transport techniques, and our results verify an intimate connection among information theory, optimal transport, and concentration of measure.
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