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Kernel Stein discrepancy (KSD) is a widely used kernel-based measure ofdiscrepancy between probability measures. It is often employed in the scenariowhere a user has a collection of samples from a candidate probability measureand wishes to compare them against a specified target probability measure. KSDhas been employed in a range of settings including goodness-of-fit testing,parametric inference, MCMC output assessment and generative modelling. However,so far the method has been restricted to finite-dimensional data. We providethe first analysis of KSD in the generality of data lying in a separableHilbert space, for example functional data. The main result is a novel Fourierrepresentation of KSD obtained by combining the theory of measure equationswith kernel methods. This allows us to prove that KSD can separate measures andthus is valid to use in practice. Additionally, our results improve theinterpretability of KSD by decoupling the effect of the kernel and Steinoperator. We demonstrate the efficacy of the proposed methodology by performinggoodness-of-fit tests for various Gaussian and non-Gaussian functional modelsin a number of synthetic data experiments.

A Fourier representation of kernel Stein discrepancy with application to Goodness-of-Fit tests for measures on infinite dimensional Hilbert spaces