Introduces tangential Bayes denoiser for Riemannian Gaussian mixtures on manifolds via spectral Laplace-Beltrami approximation, with nearly Bayes risk in low noise and minimax optimality on the circle.
Fr\'echet Means in Infinite Dimensions
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abstract
While there exists a well-developed asymptotic theory of Fr\'echet means of random variables taking values in a general "finite-dimensional" metric space, there are only a few known results in which the random variables can take values in an "infinite-dimensional" metric space. Presently, we develop a general asymptotic theory of Fr\'echet means in some infinite-dimensional metric spaces, which allows us to recover, strengthen, and generalize most existing results; in particular, we develop novel asymptotic theory for Fr\'echet means in some infinite-dimensional metric spaces from statistical shape analysis. The core of the proof is a novel notion of weak convergence in general metric spaces for which the results can be proven via calculus of variations.
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Nonparametric Riemannian Empirical Bayes, and Denoising Measurements on Manifolds
Introduces tangential Bayes denoiser for Riemannian Gaussian mixtures on manifolds via spectral Laplace-Beltrami approximation, with nearly Bayes risk in low noise and minimax optimality on the circle.