Brascamp-Lieb, Subadditivity of Entropy and a Log-Sobolev inequalitySpeaker: Juspreet
Title: Brascamp-Lieb, Subadditivity of Entropy and a Log-Sobolev inequality
Date: 12 Aug 2019 5:30pm-7:00pm
Location: Maxwell-Dworkin 323
Audience background: Probability, first-half of the first talk
Abstract: We will continue by briefly restating the important results from the first part of the last talk, and going through again (in detail) the proof for the weak form of subadditivity of entropy. We will then give motivation for the Brascamp-Lieb inequality, state it in its full generality, and develop an intuition for it through various examples (special cases). After defining these 2 notions, we will state the definition of a Fenchel-Legendre Transform of a convex function, along with the moment-generating function of a probability distribution. With these tools, we will sketch a proof that shows the duality between a (strong) form of subadditvity of entropy and the brascamp-lieb inequality using the fl-transform on the moment-generating function of the underlying distribution. We will discuss the geometric and isoperimetric ramifications of this, including a brief statement of the isoperimetric inequality for the hypercube (but with biases on the vertices). Time permitting, we will then write down an elementary log-sobolev inequality (sans proof details), and hint at its connections to a familiar inequality, and then state an exponential concentration inequality for boolean functions (implied by Log-Sobolev).