Sampling Over Riemannian Manifolds Using Kernel Herding

Sandesh Adhikary, Byron Boots

Abstract:

Kernel herding is a deterministic sampling algorithm designed to draw ‘super samples’ from probability distributions when provided with their kernel mean embeddings in a reproducing kernel Hilbert space (RKHS). Empirical expectations of functions in the RKHS formed using these super samples tend to converge even faster than random sampling from the true distribution itself. Standard implementations of kernel herding have been restricted to sampling over flat Euclidean spaces, which is not ideal for applications such as robotics where more general Riemannian manifolds may be appropriate. We propose to adapt kernel herding to Riemannian manifolds by (1) using geometry-aware kernels that incorporate the appropriate distance metric for the manifold and (2) using Riemannian optimization to constrain herded samples to lie on the manifold. We evaluate our approach on problems involving various manifolds commonly used in robotics including the SO(3) manifold of rotation matrices, the spherical manifold used to encode unit quaternions, and the manifold of symmetric positive definite matrices. We demonstrate that our approach outperforms existing alternatives on the task of resampling from empirical distributions of weighted particles, a problem encountered in applications such as particle filtering. We also demonstrate how Riemannian kernel herding can be used as part of the kernel recursive approximate Bayesian computation algorithm to estimate parameters of black-box simulators, including inertia matrices of an Adroit robot hand simulator. Our results confirm that exploiting geometric information through our approach to kernel herding yields better results than alternatives including standard kernel herding with heuristic projections.