arXiv subjects

Mathias Højgaard Jensen

Publications and source records attributed to Mathias Højgaard Jensen.

6 recordsLinked to original sources

arXiv2023

Simulation of Conditioned Semimartingales on Riemannian Manifolds

We present a scheme for simulating conditioned semimartingales taking values in Riemannian manifolds. Extending the guided bridge proposal approach used for simulating Euclidean bridges, the scheme replaces the drift of the conditioned process with an approximation in terms of a scaled radial vector field. This handles the fact that transition densities are generally intractable on geometric spaces. We prove the validity of the scheme by a change of measure argument, and we show how the resulting guided processes can be used in importance sampling and for approximating the density of the unconditioned process. The scheme is used for numerically simulating bridges on two- and three-dimensional manifolds, for approximating otherwise intractable transition densities, and for estimating the diffusion mean of sampled geometric data.

math.NA↗

arXiv2022

Diffusion within pores fully revealed by magnetic resonance

Probing the transport of fluids within confined domains is important in many areas including material science, catalysis, food science, and cell biology. The diffusion propagator fully characterizes the diffusion process, which is highly sensitive to the confining boundaries as well as the structure within enclosed pores. While magnetic resonance has been used extensively to observe various features of the diffusion process, its full characterization has been elusive. Here, we address this challenge by employing a special sequence of magnetic field gradient pulses for measuring the diffusion propagator, which allows for `listening to the drum' and determining not only the pore's shape but also diffusive dynamics within it.

cond-mat.mtrl-sci↗

arXiv2022

Bridge Simulation and Metric Estimation on Lie Groups and Homogeneous Spaces

We present schemes for simulating Brownian bridges on complete and connected Lie groups and homogeneous spaces. We use this to construct an estimation scheme for recovering an unknown left- or right-invariant Riemannian metric on the Lie group from samples. We subsequently show how pushing forward the distributions generated by Brownian motions on the group results in distributions on homogeneous spaces that exhibit non-trivial covariance structure. The pushforward measure gives rise to new parametric families of distributions on commonly occurring spaces such as spheres and symmetric positive tensors. We extend the estimation scheme to fit these distributions to homogeneous space-valued data. We demonstrate both the simulation schemes and estimation procedures on Lie groups and homogenous spaces, including $\SPD(3) = \GL_+(3)/\SO(3)$ and $\mathbb S^2 = \SO(3)/\SO(2)$.

stat.CO↗

arXiv2022

Diffusion Mean Estimation on the Diagonal of Product Manifolds

Computing sample means on Riemannian manifolds is typically computationally costly as exemplified by computation of the Fréchet mean which often requires finding minimizing geodesics to each data point for each step of an iterative optimization scheme. When closed-form expressions for geodesics are not available, this leads to a nested optimization problem that is costly to solve. The implied computational cost impacts applications in both geometric statistics and in geometric deep learning. The weighted diffusion mean offers an alternative to the weighted Fréchet mean. We show how the diffusion mean and the weighted diffusion mean can be estimated with a stochastic simulation scheme that does not require nested optimization. We achieve this by conditioning a Brownian motion in a product manifold to hit the diagonal at a predetermined time. We develop the theoretical foundation for the sampling-based mean estimation, we develop two simulation schemes, and we demonstrate the applicability of the method with examples of sampled means on two manifolds.

stat.ME↗

arXiv2021

Bridge Simulation and Metric Estimation on Lie Groups

We present a simulation scheme for simulating Brownian bridges on complete and connected Lie groups. We show how this simulation scheme leads to absolute continuity of the Brownian bridge measure with respect to the guided process measure. This result generalizes the Euclidean result of Delyon and Hu to Lie groups. We present numerical results of the guided process in the Lie group $\SO(3)$. In particular, we apply importance sampling to estimate the metric on $\SO(3)$ using an iterative maximum likelihood method.

math.ST↗

arXiv2019

Simulation of Conditioned Diffusions on the Flat Torus

Diffusion processes are fundamental in modelling stochastic dynamics in natural sciences. Recently, simulating such processes on complicated geometries has found applications for example in biology, where toroidal data arises naturally when studying the backbone of protein sequences, creating a demand for efficient sampling methods. In this paper, we propose a method for simulating diffusions on the flat torus, conditioned on hitting a terminal point after a fixed time, by considering a diffusion process in R 2 which we project onto the torus. We contribute a convergence result for this diffusion process, translating into convergence of the projected process to the terminal point on the torus. We also show that under a suitable change of measure, the Euclidean diffusion is locally a Brownian motion.

math.PR↗