Discrete-Time Observations of Brownian Motion on Lie Groups and Homogeneous Spaces: Sampling and Metric Estimation

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 t...

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Bibliographic Details
Published in:Algorithms 2022-08, Vol.15 (8), p.290
Main Authors: Jensen, Mathias Højgaard, Joshi, Sarang, Sommer, Stefan
Format: Article
Language:English
Subjects:
bridge simulation
Brownian motion
Diffusion
directional statistics
Euclidean space
homogeneous spaces
Lie groups
metric estimation
Metric spaces
Normal distribution
Parameter estimation
Simulation
Tensors
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Summary: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 a non-trivial covariance structure. The pushforward measure gives rise to new non-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 S2=SO(3)/SO(2).
ISSN:1999-4893
1999-4893
DOI:10.3390/a15080290