Title: Riemannian Proximal Sampler for High-accuracy Sampling on Manifolds

 

Abstract:​ 

 

Sampling from densities ​defined on Riemannian manifolds is central to Bayesian inference, generative modeling, and differential privacy. We introduce the Riemannian Proximal Sampler (RPS), whose efficiency hinges on two oracles: Manifold Brownian Increments and the Riemannian Heat Kernel. We establish high-accuracy sampling guarantees for the Riemannian Proximal Sampler, showing that generating samples with ε-accuracy requires O(log(1/ε)) iterations in Kullback-Leibler divergence assuming access to exact oracles and O(log​^2(1/ε)) iterations in the total variation metric assuming access to sufficiently accurate inexact oracles. Furthermore, we present two practical implementations of these oracles by leveraging heat-kernel truncation and Varadhan's asymptotics, respectively. In the latter case, we interpret the Riemannian Proximal Sampler as a discretization of the entropy-regularized Riemannian Proximal Point Method on the associated Wasserstein space. We will discuss numerical results that illustrate the effectiveness of the proposed methodology.

 

​Bio: 

 

Krishnakumar Balasubramanian is an Associate Professor in the Department of Statistics at the University of California, Davis, affiliated with the Graduate Group in Applied Mathematics, the Center for Data Science and Artificial Intelligence Research (CeDAR), and the TETRAPODS Institute of Data Science. He is also an Amazon Scholar and was a visiting scientist at the Simons Institute for the Theory of Computing at UC Berkeley in Fall 2021 and Fall 2022. Krishna received his Ph.D. in Computer Science from the Georgia Institute of Technology and completed postdoctoral research at Princeton University and the University of Wisconsin–Madison. His research lies at the interface of machine learning and artificial intelligence, statistics and optimization. He is a recipient of several honors, including a Facebook Fellowship (2013), the ICML Best Paper Runner-Up Award (2013), and the INFORMS ICS Prize (2024). He contributes actively to the academic community as an Associate Editor for the Annals of Statistics, IEEE Transactions on Information Theory and the Journal of Machine Learning Research, and serves regularly as a (senior) area chair for leading conferences such as ICML, ICLR, NeurIPS, and COLT.