The Concentration of Measure Phenomenon

We cannot find n orthogonal vectors in Euclidean space unless the dimension is larger than or equal to n. If we are willing to settle for n vectors that are almost orthogonal, however, we can even find those in a Euclidean space of dimension ∼ log(n).
This drastic reduction in the dimension of the ambient space is due to a very general phenomenon which occurs in many high-dimensional spaces, and implies that any function with small local variations, is actually almost constant on a very large proportion of the space.
The so-called concentration of measure phenomenon is a powerful tool with applications in different fields of mathematics. Many asymptotic results in probability theory, information theory and dynamical systems such as the law of large numbers, the asymptotic equipartition property, the central limit theorem and Birkhoff's theorem are consequences of concentration of measure.
We will discuss various methods for deriving concentration properties, discuss examples and applications. We will go over isoperimetric inequalities, geometric and topological aspects, concentration in product spaces, logarithmic Sobolev inequalities, the relation to curvature, transportation costs and relative entropy (Kullback-Leibler divergence).

Date and time info
Thursday 11.00 - 12.30

Keywords
Concentration of measure, asymptotic geometric analysis (highdimensional Banach spaces), Ricci curvature

Prerequisites
Multivariate calculus; probability or measure theory

Audience
Diploma students, MSc students, PhD students, Postdocs

Language
English