Score matching for sub-Riemannian bridge sampling
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910419603423232 |
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| author | Grong, Erlend Habermann, Karen Sommer, Stefan |
| author_facet | Grong, Erlend Habermann, Karen Sommer, Stefan |
| contents | Simulation of conditioned diffusion processes is an essential tool in inference for stochastic processes, data imputation, generative modelling, and geometric statistics. Whilst simulating diffusion bridge processes is already difficult on Euclidean spaces, when considering diffusion processes on Riemannian manifolds the geometry brings in further complications. In even higher generality, advancing from Riemannian to sub-Riemannian geometries introduces hypoellipticity, and the possibility of finding appropriate explicit approximations for the score of the diffusion process is removed. We handle these challenges and construct a method for bridge simulation on sub-Riemannian manifolds by demonstrating how recent progress in machine learning can be modified to allow for training of score approximators on sub-Riemannian manifolds. Since gradients dependent on the horizontal distribution, we generalise the usual notion of denoising loss to work with non-holonomic frames using a stochastic Taylor expansion, and we demonstrate the resulting scheme both explicitly on the Heisenberg group and more generally using adapted coordinates. We perform numerical experiments exemplifying samples from the bridge process on the Heisenberg group and the concentration of this process for small time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_15258 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Score matching for sub-Riemannian bridge sampling Grong, Erlend Habermann, Karen Sommer, Stefan Probability Machine Learning Differential Geometry Statistics Theory 58J65, 53C17, 62R30 Simulation of conditioned diffusion processes is an essential tool in inference for stochastic processes, data imputation, generative modelling, and geometric statistics. Whilst simulating diffusion bridge processes is already difficult on Euclidean spaces, when considering diffusion processes on Riemannian manifolds the geometry brings in further complications. In even higher generality, advancing from Riemannian to sub-Riemannian geometries introduces hypoellipticity, and the possibility of finding appropriate explicit approximations for the score of the diffusion process is removed. We handle these challenges and construct a method for bridge simulation on sub-Riemannian manifolds by demonstrating how recent progress in machine learning can be modified to allow for training of score approximators on sub-Riemannian manifolds. Since gradients dependent on the horizontal distribution, we generalise the usual notion of denoising loss to work with non-holonomic frames using a stochastic Taylor expansion, and we demonstrate the resulting scheme both explicitly on the Heisenberg group and more generally using adapted coordinates. We perform numerical experiments exemplifying samples from the bridge process on the Heisenberg group and the concentration of this process for small time. |
| title | Score matching for sub-Riemannian bridge sampling |
| topic | Probability Machine Learning Differential Geometry Statistics Theory 58J65, 53C17, 62R30 |
| url | https://arxiv.org/abs/2404.15258 |