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A unsupervised learning approach for the computation of an explicitfunctional representation of a random vector $Y$ is presented, which onlyrelies on a finite set of samples with unknown distribution. Motivated byrecent advances with computational optimal transport for estimating Wassersteindistances, we develop a new \textit{Wasserstein multi-element polynomial chaosexpansion} (WPCE). It relies on the minimization of a regularized empiricalWasserstein metric known as debiased Sinkhorn divergence.  As a requirement for an efficient polynomial basis expansion, a suitable(minimal) stochastic coordinate system $X$ has to be determined with the aim toidentify ideally independent random variables. This approach generalizesrepresentations through diffeomorphic transport maps to the case ofnon-continuous and non-injective model classes $\mathcal{M}$ with differentinput and output dimension, yielding the relation $Y=\mathcal{M}(X)$ indistribution. Moreover, since the used PCE grows exponentially in the number ofrandom coordinates of $X$, we introduce an appropriate low-rank format given asstacks of tensor trains, which alleviates the curse of dimensionality, leadingto only linear dependence on the input dimension. By the choice of the modelclass $\mathcal{M}$ and the smooth loss function, higher order optimizationschemes become possible. It is shown that the relaxation to a discontinuousmodel class is necessary to explain multimodal distributions. Moreover, theproposed framework is applied to a numerical upscaling task, considering acomputationally challenging microscopic random non-periodic composite material.This leads to tractable effective macroscopic random field in adoptedstochastic coordinates.

Low-rank Wasserstein polynomial chaos expansions in the framework of optimal transport