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Probabilistic learning on manifolds

Abstract : This paper presents novel mathematical results in support of the probabilistic learning on manifolds (PLoM) recently introduced by the authors. An initial dataset, constituted of a small number of points given in an Euclidean space, is given. The points are independent realizations of a vector-valued random variable for which its non-Gaussian probability measure is unknown but is, a priori, concentrated in an unknown subset of the Euclidean space. A learned dataset, constituted of additional realizations, is constructed. A transport of the probability measure estimated with the initial dataset is done through a linear transformation constructed using a reduced-order diffusion-maps basis. It is proven that this transported measure is a marginal distribution of the invariant measure of a reduced-order Itô stochastic differential equation. The concentration of the probability measure is preserved. This property is shown by analyzing a distance between the random matrix constructed with the PLoM and the matrix representing the initial dataset, as a function of the dimension of the basis. It is further proven that this distance has a minimum for a dimension of the reduced-order diffusion-maps basis that is strictly smaller than the number of points in the initial dataset.
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Submitted on : Friday, August 21, 2020 - 4:07:36 PM
Last modification on : Sunday, June 26, 2022 - 2:52:45 AM
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Christian Soize, Roger Ghanem. Probabilistic learning on manifolds. Foundations of Data Science, American Institute of Mathematical Sciences, 2020, 2 (3), pp.279-307. ⟨10.3934/fods.2020013⟩. ⟨hal-02919127⟩



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