Wasserstein Dictionary Learning: Optimal Transport-based unsupervised non-linear dictionary learning

摘要

This article introduces a new non-linear dictionary learning method forhistograms in the probability simplex. The method leverages optimal transporttheory, in the sense that our aim is to reconstruct histograms using so calleddisplacement interpolations (a.k.a. Wasserstein barycenters) between dictionaryatoms; such atoms are themselves synthetic histograms in the probabilitysimplex. Our method simultaneously estimates such atoms, and, for eachdatapoint, the vector of weights that can optimally reconstruct it as anoptimal transport barycenter of such atoms. Our method is computationallytractable thanks to the addition of an entropic regularization to the usualoptimal transportation problem, leading to an approximation scheme that isefficient, parallel and simple to differentiate. Both atoms and weights arelearned using a gradient-based descent method. Gradients are obtained byautomatic differentiation of the generalized Sinkhorn iterations that yieldbarycenters with entropic smoothing. Because of its formulation relying onWasserstein barycenters instead of the usual matrix product between dictionaryand codes, our method allows for non-linear relationships between atoms and thereconstruction of input data. We illustrate its application in severaldifferent image processing settings.