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Probabilistic learning on manifolds

  • * Corresponding author: Christian Soize

    * Corresponding author: Christian Soize 
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  • 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.

    Mathematics Subject Classification: Primary: 68Q32, 62G09; Secondary: 60J22.


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  • Figure 1.  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) $

    Figure 2.  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 $

    Figure 3.  Left figure: graph of function $ m \mapsto f_d(m) $. Right figure: graph of function $ m\mapsto \underline g(m) $

    Figure 4.  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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