講演者

Hui-An Shen (Institut Polytechnique de Paris)

会場

埼玉大学 理学部1号館3階 基礎数理演習室

アブストラクト

The overall goal of this talk is to introduce geometric structures that remain invariant under stochastic maps that are, roughly speaking, non-mixing.
First, I will introduce the geometry of uncoded transmission, concerning the well‑known example of transmitting the Gaussian source over the Gaussian channel and its unique optimality in point-to-point communication under certain constraints, using convex bodies in (real) normed linear spaces. I will show how a homothetic property of convex bodies accounts for the uniqueness of Gaussian uncoded transmission among continuous sources and additive channels (with symmetric, continuous, log-concave distributions). I will argue how the said additive channels are the continuous analog of Markov embeddings for discrete probability measures, in that both capture the non-mixing property.
Next, I will briefly introduce Chentsov’s Theorem, foundational in information geometry, which characterizes the Riemannian metrics satisfying invariance under Markov embeddings.
In conclusion, I will explain how the geometry of uncoded transmission is essentially about invariant norms in the space of random sequences, while Chentsov’s Theorem is about invariant inner products in the space of discrete probability measures.