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Start date: 2022 × End date: 2022 × Start date: 2020 × DDC: 310 × Version: publishedVersion × WGL Institute: WIAS ×

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    Gaussian processes with multidimensional distribution inputs via optimal transport and Hilbertian embedding
    (Ithaca, NY : Cornell University Library, 2020) Bachoc, François; Suvorikova, Alexandra; Ginsbourger, David; Loubes, Jean-Michel; Spokoiny, Vladimir
    In this work, we propose a way to construct Gaussian processes indexed by multidimensional distributions. More precisely, we tackle the problem of defining positive definite kernels between multivariate distributions via notions of optimal transport and appealing to Hilbert space embeddings. Besides presenting a characterization of radial positive definite and strictly positive definite kernels on general Hilbert spaces, we investigate the statistical properties of our theoretical and empirical kernels, focusing in particular on consistency as well as the special case of Gaussian distributions. A wide set of applications is presented, both using simulations and implementation with real data.
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    Convergence analysis of Tikhonov regularization for non-linear statistical inverse problems
    (Ithaca, NY : Cornell University Library, 2020) Rastogi, Abhishake; Blanchard, Gilles; Mathé, Peter
    We study a non-linear statistical inverse problem, where we observe the noisy image of a quantity through a non-linear operator at some random design points. We consider the widely used Tikhonov regularization (or method of regularization) approach to estimate the quantity for the non-linear ill-posed inverse problem. The estimator is defined as the minimizer of a Tikhonov functional, which is the sum of a data misfit term and a quadratic penalty term. We develop a theoretical analysis for the minimizer of the Tikhonov regularization scheme using the concept of reproducing kernel Hilbert spaces. We discuss optimal rates of convergence for the proposed scheme, uniformly over classes of admissible solutions, defined through appropriate source conditions.