Computable Statistical Divergences for Functional Data

Kernel-based discrepancies have found considerable success in constructing statistical tests which are now widely used in statistical machine learning.  Examples include Kernel Stein Discrepancy which enables goodness-of-fit tests of data samples against an (unnormalized)  probability density based on Stein's method.  The effectiveness of the associated tests will crucially depend on the dimension of the data.
I will present some recent results on the behaviour of such tests in high dimensions,  exploring properties of the statistical divergence under different scaling of data dimension and data size.
Building on this, I will discuss how such discrepancies can be extended to probability distributions on infinite-dimensional spaces.  I will discuss applications to goodness-of-fit testing for measures on function spaces and its relevance to various problems in uncertainty quantification.