These slides contain new results about stochastic approximation with operator means. First I report on new results related to generalized Karcher means and law of large numbers for such. This is a continuation of earlier work on the subject in Thompson metric spaces. Then new results on Wasserstein means of positive operators are reviewed, in particular their definition, construction and approximation are considered. This parallels the theory of the Karcher mean developed in Thompson metric spaces, but the underlying geometry is totally different, in particular we don’t know too much about the constructed Banach-Finsler structure that is instrumental to have exponential contracting properties of associated flows and thus law of large numbers type of results.
Miklós Pálfia 