  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.
