Learning, inference, and prediction on manifolds: constrained Gaussian processes on PDFs

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In this paper, we introduce a new framework to learn, infer and predict nonparametric PDFs with a Gaussian process prior. This is a challenging problem since there is no explicit formulas for conditional expectations and covariances. The proposed methods have two main advantages: Analyzing PDFs with an intrinsic manifold structure and establishing the connection with the Hilbert sphere when endowed with an appropriate metric. Therefore, all solutions are valid PDFs with non-negative values and integrate to one. We formulate the problem as constrained spherical Gaussian processes leading to an efficient solution with a spherical Hamiltonian Monte Carlo (HMC) sampling. We test and evaluate different strategies with extensive experiments on both simulations and real dataset.