In this paper we explore domain adaptation through optimal transport. We propose a novel approach, where we model the data distributions through Gaussian mixture models. This strategy allows us to solve continuous optimal transport through an equivalent discrete problem. The optimal transport solution gives us a matching between source and target domain mixture components. From this matching, we can map data points between domains, or transfer the labels from the source domain components towards the target domain. We experiment with 2 domain adaptation benchmarks in fault diagnosis, showing that our methods have state-of-the-art performance.

This paper introduces a new nonlinear dictionary learning method for histograms in the probability simplex. The method leverages optimal transport theory, in the sense that our aim is to reconstruct histograms using so-called displacement interpolations (a.k.a. Wasserstein barycenters) between dictionary atoms; such atoms are themselves synthetic histograms in the probability simplex. Our method simultaneously estimates such atoms, and, for each datapoint, the vector of weights that can optimally reconstruct it as an optimal transport barycenter of such atoms. Our method is computationally tractable thanks to the addition of an entropic regularization to the usual optimal transportation problem, leading to an approximation scheme that is efficient, parallel and simple to differentiate. Both atoms and weights are learned using a gradient-based descent method. Gradients are obtained by automatic differentiation of the generalized Sinkhorn iterations that yield barycenters with entropic smoothing. Because of its formulation relying on Wasserstein barycenters instead of the usual matrix product between dictionary and codes, our method allows for nonlinear relationships between atoms and the reconstruction of input data. We illustrate its application in several different image processing settings.