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Journal Articles Foundations of Data Science Year : 2020

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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Dates and versions

hal-02919127 , version 1 (21-08-2020)

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Christian Soize, Roger Ghanem. Probabilistic learning on manifolds. Foundations of Data Science, 2020, 2 (3), pp.279-307. ⟨10.3934/fods.2020013⟩. ⟨hal-02919127⟩
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