Global Fréchet Manifold Learning for Random Objects, With Application to Low-Dimensional Wasserstein  Representations of Distributional Data

Álvaro Gajardo; Hans-Georg Müller

We study manifold learning with multidimensional scaling for samples of metric space valued data. By adopting a global version of ISOMAP we obtain low-dimensional Euclidean representations. A key innovation is that we demonstrate that global Fréchet regression can be utilized for mapping the elements of a convex set in the Euclidean representation space back to the metric space where the objects reside. We refer to this approach as Fréchet manifold learning and showcase it with one-dimensional distributions as random objects, equipped with the Wasserstein metric, which is an important special case of our general approach. The resulting low-dimensional representations mimic the parametric representation in a parametric family of distributions but are entirely learned from the data without postulating any parametric model. These Wasserstein representations of distributional data can be viewed as an empirical parametrization of a sample of distributions. The utility of these representations rests on the map from the low-dimensional Euclidean representation space to the space of distributions, which is obtained with global Fréchet regression. We illustrate the proposed approach with distributional data for baby names, bike rentals and age pyramids and further demonstrate how it can be applied for a novel distributional regression method that features one-dimensional distributions as predictors.

Global Fréchet Manifold Learning for Random Objects, With Application to Low-Dimensional Wasserstein Representations of Distributional Data

Abstract