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Asymptotic Analysis Seminar: Optimal Concentration of Information for Log-Concave Distributions

Tuesday, March 3, 2015 - 11:00am - 12:00pm
Lind 305
Mokshay Madiman (University of Delaware)
Recently it was shown by Bobkov and the speaker that for a random vector X in R^n drawn from a log-concave density e^{-V}, the information content per coordinate, namely V(X)/n, is highly concentrated about its mean. Their argument was nontrivial, involving the localization technique, and also gave suboptimal exponents, but it was sufficient to demonstrate that high-dimensional log-concave measures are in a sense close to uniform distributions on the annulus between 2 nested convex sets. We will show that one can obtain an optimal concentration bound in this setting (optimal even in the constant terms, not just the exponent), using very simple techniques, and give a complete proof. We will also mention some applications, including to random matrix theory. This is joint work with Matthieu Fradelizi and Liyao Wang.