CS21: Stochastic Numerics on Manifolds

date: 7/15/2025, time: 16:00-17:30, room: ICS 118

Organizer: Alexander Lewis (Georg-August-Universität Göttingen)

Chair: Alexander Lewis (Georg-August-Universität Göttingen)

Sampling on manifolds via Langevin diffusion

Akash Sharma (Chalmers University of Technology)

We derive error bounds for sampling and estimation using a discretization of an intrinsically defined Langevin diffusion on Riemannian manifold. Imposing no restrictions beyond a nominal level of smoothness on potential function, first-order error bounds, in discretization step size, on the bias and variances of estimators are derived. We will also discuss conditions for extending analysis to the case of non-compact manifolds and different variants of the algorithm. We will present numerical illustrations with distributions on the manifolds which verify the derived bounds.

Bibliography

\([1]\) K. Bharath, A. Lewis, A. Sharma, M.V. Tretyakov "Sampling and estimation on manifolds using the Langevin diffusion" Journal of Machine Learning Research, vol. 26, (2025), pp. 1-50.

Fundamental theorem for mean square convergence of SDEs on Riemannian manifolds

Alexander Lewis (Georg-August-Universität Göttingen)

The error rate for the Euler(-Murayama) approximation of SDEs on Riemannian manifolds in the weak sense has been established in \([1]\) and was found to be of global rate 1; reflecting the classical result known Euclidean space. However, strong convergence rates of the Euler scheme have yet to be derived. Though based on intuition, it is not unreasonable to expect that the manifold scheme has the same global rate as its Euclidean counterpart of 1/2.

By following closely to the approach laid out in the seminal works of Milstein, we show how to generate high order strong schemes on a Riemannian manifold with empty cut-locus. In particular, we show that the Euler scheme has global rate 1/2. Furthermore, we present the Milstein correction to the Euler scheme which yields a scheme of global order 1. Finally, we will formulate the manifold generalisation of the fundamental theorem of strong convergence, allowing us to obtain global convergence rates for a wide range of numerical schemes. If time permits, I will also present numerical experiments which illustrate the theoretical guarantees.

The talk will give an overview of results obtained in joint work with Karthik Bharath and Michael Tretyakov.

\([1]\) K. Bharath, A. Lewis, A. Sharma, M.V. Tretyakov "Sampling and estimation on manifolds using the Langevin diffusion" Journal of Machine Learning Research, vol. 26, (2025).

Kinetic Langevin equations on Lie groups with a geometric mechanics approach

Erwin Luesink (University of Amsterdam)

Kinetic Langevin equations are used for sampling from distributions and can employed in spaces with nontrivial curvature. On Lie groups, these equations can be solved with efficient numerical algorithms. In this talk, we discuss how the kinetic Langevin equations can be viewed within the framework of stochastic geometric mechanics. The benefit of doing so is that one obtains a physical interpretation as well as a multiplicative noise version of the kinetic Langevin equations.