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Abstract:
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In nonparametric regression and spatial process modeling, it is common for the inputs to fall in a restricted subset of Euclidean space. Typical kernel-based methods that do not take into account the intrinsic geometric of the domain across which observations are collected may produce sub-optimal results. We focus on solving this problem in the context of Gaussian process (GP) models, proposing a new class of diffusion-based GPs (DB-GPs), which learn a covariance that respects the geometry of the input domain. We use the term `diffusion-based' as the idea is to measure intrinsic distances between inputs in a restricted domain via a diffusion process. As the heat kernel is intractable computationally, we approximate the covariance using finitely-many eigenpairs of the Graph Laplacian (GL). The theory for the DB-GP methodology will be introduced in the talk. Finally, the performance of DB-GP on toy examples, simulation studies, and applications to ecology data will be illustrated.
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