Clément Sarrazin - Statistical spectral analysis of Transfer Operators via Entropic Optimal Transport

In this talk, we will interest ourselves with the inference of singular patterns (cyclical/invariant sets, clusters of shared behaviours,…) in stochastic dynamical systems, from empirical observations of its transitions. In theory, the associated L² "transfer operator" yields a very elegant way to isolate some of these paterns, via its spectral or singular values decomposition. In practice however, this infinite dimensional object is often untractable in most of its aspects and, in some applications, not even known in closed form. Furthermore, the finer parts of its spectrum might not even reflect meaningful behaviors for most applications.

We will introduce an (artificial) observation model for the empirical data which allows to define an empirical version of the transfer operator, using structures borrowed from the theory of entropic optimal transport. These finite dimensional "entropic transfer operators" allow to recover the spectral/singular information of the sampled dynamic, as more and more data is observed. It is also, in general, much simpler to decompose and analyse, while also allowing some selection in the scale at which the spectral information is reconstructed , through the entropic bluring of optimal transport.

This is part of a joint work with  H. Bi and B. Schmitzer of Göttingen university.