Abstract
We theoretically analyze the properties of a geodesic random walk on the Euclidean . Specifically, we prove that the random walk’s transition kernel is Wasserstein contractive with a contraction rate which can be bounded from above independently of the dimension d. Our result is of particular interest due to its implications regarding the potential for dimension-independent performance of both geodesic slice sampling on the sphere and Gibbsian polar slice sampling, which are Markov chain Monte Carlo methods for approximate sampling from essentially arbitrary distributions on their respective state spaces.
Funding Statement
PS gratefully acknowledges funding by the Carl Zeiss Stiftung within the program “CZS Stiftungsprofessuren” and the project “Interactive Inference”. Moreover, PS is thankful for support by the DFG within project 432680300 – Collaborative Research Center 1456 “Mathematics of Experiment”.
Acknowledgments
We are thankful for the helpful feedback of an anonymous referee. We thank Björn Sprungk for the discussion that led to our investigation, in particular for drawing our attention to the question of dimension dependence of GSSS for a constant target, in the context of its relevance for GPSS. We thank Mareike Hasenpflug for discussions on the topic and helpful comments on a preliminary version of the manuscript.
Citation
Philip Schär. Thilo D. Stier. "A dimension-independent bound on the Wasserstein contraction rate of a geodesic random walk on the sphere." Electron. Commun. Probab. 29 1 - 11, 2024. https://doi.org/10.1214/24-ECP631
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