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Abstract:
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In many applications, data and/or parameters are supported on non-Euclidean spaces. It is important to take into account the geometric structure of manifolds in statistical analysis to avoid misleading results. In this talk, we consider a very broad class of manifolds: non-compact Riemannian symmetric spaces. For this class, we provide statistical models on the tangent space, push these models forward onto the manifold, and easily calculate induced distributions by Jacobians. To illustrate the statistical utility of this theoretical result, we provide a general method to construct distributions on symmetric spaces, including the log-Gaussian distribution as an analogue of the multivariate Gaussian distribution in Euclidean space. With these new kernels on symmetric spaces, any existing density estimation approach designed for Euclidean spaces can be applied, and pushed forward to the manifold with an easy-to-calculate adjustment. We provide theorems showing that the induced density estimators on the manifold inherit the statistical optimality properties of the parent Euclidean density estimator; this holds for both frequentist and Bayesian nonparametric methods.
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