A rigorous extension of the kernel Stein discrepancy - a normalization-free measure of how well a sample matches a target distribution - from flat Euclidean space to curved Riemannian manifolds. Students learn how the Stein operator is generalized consistently to a manifold, why the resulting discrepancy still separates distributions, and how the minimum-KSD estimator behaves asymptotically and performs in goodness-of-fit testing, with closed-form results on the sphere, Grassmann, Stiefel, and SPD-matrix manifolds where normalization constants are intractable.
This paper extends the kernel Stein discrepancy (KSD) - a way of measuring how far a sample is from a target probability distribution without ever computing that distribution's normalizing constant -…