- Marco Cuturi (Kyoto University – Japan) –
**Regularized Optimal Transport and Applications** - Jérémie Bigot (Université de Bordeaux et Institut de Mathématiques de Bordeaux – France) –
**A Forward-Backward algorithm for geodesic PCA of histograms in the Wasserstein space** - Rémi Flamary (Université de Nice Sophia-Antipolis – France) –
**Optimal transport for domain adaptation**

Optimal transport (OT) theory provides geometric tools to compare probability measures. After reviewing the basics of OT distances (a.k.a Wasserstein or Earth Mover’s), I will show how an adequate regularization of the OT problem can result in substantially faster (GPU parallel) and much better behaved (strongly convex) numerical computations. I will then show how this regularization can enable several applications of OT to learn from probability measures. I will focus on in particular on the computation of Wasserstein barycenters and inverse problem (regression) in the simplex with the OT geometry (the latter being joint work with G. Peyré and N. Bonneel).

Principal Component Analysis (PCA) in a linear space is certainly the most widely used approach in multivariate statistics to summarize efficiently the information in a data set. In this talk, we are concerned by the statistical analysis of data sets whose elements are histograms with support on the real line. For the purpose of dimension reduction and data visualization of variables in the space of histograms, it is of interest to compute their principal modes of variation around a mean element. However, since the number, size or locations of significant bins may vary from one histogram to another, using PCA in an Euclidean space is not an appropriate tool. In this work, an histogram is modeled as a probability density function (pdf) with support included in an interval of the real line, and the Wasserstein metric is used to measure the distance between two histograms. In this setting, the variability in a set of histograms can be analyzed via the notion of Geodesic PCA (GPCA) of probability measures in the Wasserstein space. However, the implementation of GPCA for data analysis remains a challenging task even in the simplest case of pdf supported on the real line. The main purpose of this talk is thus to present a fast algorithm which performs an exact GPCA of pdf with support on the real line, and to show its usefulness for the statistical analysis of histograms of surnames over years in France.

Domain adaptation is one of the most challenging tasks of modern data analytics. If the adaptation is done correctly, models built on a specific data representations become more robust when confronted to data depicting the same semantic concepts (the classes), but observed by another observation system with its own specificities. Among the many strategies proposed to adapt a domain to another, finding domain-invariant representations has shown excellent properties, as a single classifier can use labelled samples from the source domain under this representation to predict the unlabelled samples of the target domain. In this paper, we propose a regularized unsupervised optimal transportation model to perform the alignment of the representations in the source and target domains. We learn a transportation plan matching both PDFs, which constrains labelled samples in the source domain to remain close during transport. This way, we exploit at the same time the few labeled information in the source and distributions of the input/observation variables observed in both domains. Experiments in toy and challenging real visual adaptation examples show the interest of the method, that consistently outperforms state of the art approaches.