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Wasserstein Discriminant Analysis

Rémi Flamary 1, 2, * Marco Cuturi 3 Nicolas Courty 4 Alain Rakotomamonjy 5
* Corresponding author
4 OBELIX - Environment observation with complex imagery
IRISA-D5 - SIGNAUX ET IMAGES NUMÉRIQUES, ROBOTIQUE, UBS - Université de Bretagne Sud
5 DocApp - LITIS - Equipe Apprentissage
LITIS - Laboratoire d'Informatique, de Traitement de l'Information et des Systèmes
Abstract : Wasserstein Discriminant Analysis (WDA) is a new supervised method that canimprove classification of high-dimensional data by computing a suitable linearmap onto a lower dimensional subspace. Following the blueprint of classical Lin-ear Discriminant Analysis (LDA), WDA selects the projection matrix that maxi-mizes the ratio of two quantities: the dispersion of projected points coming fromdifferent classes, divided by the dispersion of projected points coming from thesame class. To quantify dispersion, WDA uses regularized Wasserstein distances,rather than cross-variance measures which have been usually considered, notablyin LDA. Thanks to the the underlying principles of optimal transport, WDA is ableto capture both global (at distribution scale) and local (at samples scale) interac-tions between classes. Regularized Wasserstein distances can be computed usingthe Sinkhorn matrix scaling algorithm; We show that the optimization of WDAcan be tackled using automatic differentiation of Sinkhorn iterations. Numericalexperiments show promising results both in terms of prediction and visualizationon toy examples and real life datasets such as MNIST and on deep features ob-tained from a subset of the Caltech dataset.
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Rémi Flamary, Marco Cuturi, Nicolas Courty, Alain Rakotomamonjy. Wasserstein Discriminant Analysis. Machine Learning, Springer Verlag, 2018, 107 (12), pp.1923-1945. ⟨10.1007/s10994-018-5717-1⟩. ⟨hal-01377528⟩

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