PhD defence by Mathias Højgaard Jensen

Title

Stochastic Bridge Sampling on Manifolds

Abstract

This dissertation studies simulation of guided bridge processes on Riemannian manifolds, Lie groups, and homogeneous spaces. In the first paper of the thesis, we present and evaluate a bridge simulation scheme on general Riemannian manifolds used to obtain approximations of the transition densities. In the second paper, we utilize the simulation scheme developed in the first paper to derive a simulation scheme on Lie groups and homogeneous spaces. An iterative maximum likelihood procedure applied to the approximated transition density yields an estimate of the metric structure on Lie groups. Moreover, we obtain non-isotropic distributions on homogeneous spaces, arising from non-invariant metrics on the total space. In the third and last paper of the thesis, we consider bridge simulations on a Riemannian product manifold. Given a set of data points on a manifold, we obtain an estimate of the diffusion mean by simulating a single diffusion bridge on the product manifold conditioned on the diagonal.

Supervisors

Principal Supervisor Stefan Horst Sommer

Assessment Committee

Professor Jon Sporring, DIKU
Professor Francois-Xavier Vialard, Université Gustave Eiffel
Associate Professor Frank van der Meulen, TU Delft

Moderator of defence: Associate Professor François Lauze, DIKU

For a digital copy of the thesis, please visit https://di.ku.dk/english/research/phd/.