Georgia Institute of Technology College of Sciences

The problem of quantifying the amount of information loss due to a random transformation (or a noisy channel) arises in a variety of contexts, such as machine learning, stochastic simulation, error-correcting codes, or computation in circuits with noisy gates, to name just a few. This talk will focus on discrete channels, where both the input and output sets are finite. The noisiness of a discrete channel can be measured by comparing suitable functionals of the input and output distributions. For instance, if we fix a reference input distribution, then the worst-case ratio of output relative entropy (Kullback-Leibler divergence) to input relative entropy for any other input distribution is bounded by one, by the data processing theorem. However, for a fixed reference input distribution, this quantity may be strictly smaller than one, giving so-called strong data processing inequalities (SDPIs). I will show that the problem of determining both the best constant in an SDPI and any input distributions that achieve it can be addressed using logarithmic Sobolev inequalities, which relate input relative entropy to certain measures of input-output correlation. I will also show that SDPIs for Kullback-Leibler divergence arises as a limiting case of a family of SDPIs for Renyi divergence, and discuss the relationship to hypercontraction of Markov operators.

Emory University, the Georgia Institute of Technology and Georgia State University, with support from the National Security Agency and the National Science Foundation, are hosting a series of mini-conferences. The ninth in the series will be held at Georgia Tech on April 27-28, 2013. This mini-conference's featured speaker is Dr. Fan Chung Graham, who will give two one-hour lectures. There will be five one-hour talks and a number of half-hour talks given by other invited speakers. To register, please submit the registration form. Registration is free but is required.

In this series of talks I will begin by discussing the idea of studying smooth manifolds and their submanifolds using the symplectic (and contact) geometry of their cotangent bundles. I will then discuss Legendrian contact homology, a powerful invariant of Legendrian submanifolds of contact manifolds. After discussing the theory of contact homology, examples and useful computational techniques, I will combine this with the conormal discussion to define Knot Contact Homology and discuss its many wonders properties and conjectures concerning its connection to other invariants of knots in S^3.