RM+ML: 22. Resolvent Method and Cumulant Expansion

The lecture notes for the course can be found at https://rolandspeicher.com/wp-content...
resolvent method, characteristic function, cumulant, set partition

0:00 How to generalize Stein's identity
12:27 Motivation for cumulant expansion
23:55 One-dimensional cumulants and cumulant expansion
32:35 Multivariate cumulants
39:07 Multivariate cumulant expansion
46:21 Relation between moments and cumulants

The goal of this lecture series is to cover mathematical interesting aspects of neural networks, in particular, those related to random matrices. In this 22th lecture we generalize Stein's lemma from the last lecture to non-Gaussian and non-independent situations, leading to the so-called resolvent method. This gives an expansion in so-called cumulants. Those will be motivated and described both in analytic and combinatorial terms.