NDLI: Wasserstein Dictionary Learning: Optimal Transport-Based Unsupervised Nonlinear Dictionary Learning

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Heitz, Matthieu Bonneel, Nicolas Cuturi, Marco Starck, Jean-Luc Coeurjolly, David Schmitz, Morgan A. Ngolè, Fred Peyré, Gabriel
Copyright Year	2018
Abstract	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.
Sponsorship	Agence Nationale de la Recherche. European Community. Centre National d Etudes Spatiales
Starting Page	643
Ending Page	678
Page Count	36
File Format	PDF
DOI	10.1137/17M1140431
e-ISSN	19364954
Journal	SIAM Journal on Imaging Sciences (SJISBI)
Issue Number	1
Volume Number	11
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2018-03-01
Access Restriction	Subscribed
Subject Keyword	Numerical approximation and evaluation Special problems of linear programming Wasserstein barycenter optimal transport dictionary learning
Content Type	Text
Resource Type	Article

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