# American Institute of Mathematical Sciences

2020, 2(3): 279-307. doi: 10.3934/fods.2020013

## Probabilistic learning on manifolds

 1 Université Gustave Eiffel, Laboratoire Modélisation et Simulation Multi-Echelle, MSME UMR 8208 CNRS, 5 bd Descartes, 77454 Marne-la-Vallée, France 2 University of Southern California, Viterbi School of Engineering, 210 KAP Hall, Los Angeles, CA 90089, USA

* Corresponding author: Christian Soize

Published   2020 Early access  August 2020

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.

Citation: Christian Soize, Roger Ghanem. Probabilistic learning on manifolds. Foundations of Data Science, 2020, 2 (3) : 279-307. doi: 10.3934/fods.2020013
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Left figure: for $\varepsilon_ {\hbox{DM}} = \varepsilon_ {\hbox{opt}}$, distribution of the eigenvalues $\lambda_\alpha(\varepsilon_ {\hbox{opt}})$ in log scale as a function of rank $\alpha$. Right figure: graph of function $m\mapsto \varepsilon_d(m)$
Left figure: distribution of the eigenvalues $\lambda_\alpha(\varepsilon_ {\hbox{opt}})$ in log scale as a function of rank $\alpha\leq 50$ for $\varepsilon_ {\hbox{DM}} = \varepsilon_ {\hbox{opt}} = 60$. Right figure: graph of function $m\mapsto \varepsilon_d(m)$ for $m\leq 50$
Left figure: graph of function $m \mapsto f_d(m)$. Right figure: graph of function $m\mapsto \underline g(m)$
Left figure: graph of function $m \mapsto d_N^{2, {\hbox{sim}}}(m)$. Right figure: graph of function $m \mapsto d_N^{2, {\hbox{sim}}}(m)$ (blue dashed line), and for $m\geq m_ {\hbox{opt}}$, graphs of $m \mapsto d_N^{2,c}(m)$ (dark thick straight line) and $m \mapsto d_N^{2, {\hbox{app}}}(m)$ (red thick curve line)
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