We describe a construction of Stein kernels using moment maps, which are solutions to a variant of the Monge–Ampère equation. As a consequence, we show how regularity bounds in certain weighted Sobolev spaces on these maps control the rate of convergence in the classical central limit theorem, and derive new rates in Kantorovitch–Wasserstein distance in the log-concave situation, with explicit polynomial dependence on the dimension.
"Stein kernels and moment maps." Ann. Probab. 47 (4) 2172 - 2185, July 2019. https://doi.org/10.1214/18-AOP1305