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Abstract:
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It is often useful to sample from distributions that are defined over Lie groups or homogeneous spaces such as spheres, hyperbolic space, Grassmannian manifolds or the space of positive definite matrices. We introduce ways of doing so by defining a manifestly global and coordinate-independent Hamiltonian motion on homogeneous spaces, which avoids the issues related to change of coordinates and coordinate singularities. We show how to construct symplectic integrators that automatically and exactly stay on the manifold, thus allowing large steps to be taken. Finally we apply our method to show how to sample efficiently from the space of positive definite matrices.
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