Wasserstein Discriminant Analysis
This method addresses classification challenges in high-dimensional data for machine learning applications, but it is incremental as it builds on classical Linear Discriminant Analysis with a new distance metric.
The paper tackles the problem of improving classification of high-dimensional data by introducing Wasserstein Discriminant Analysis (WDA), a supervised method that uses regularized Wasserstein distances to compute a linear projection for better class separation, showing promising results on datasets like MNIST and Caltech features.
Wasserstein Discriminant Analysis (WDA) is a new supervised method that can improve classification of high-dimensional data by computing a suitable linear map onto a lower dimensional subspace. Following the blueprint of classical Linear Discriminant Analysis (LDA), WDA selects the projection matrix that maximizes the ratio of two quantities: the dispersion of projected points coming from different classes, divided by the dispersion of projected points coming from the same class. To quantify dispersion, WDA uses regularized Wasserstein distances, rather than cross-variance measures which have been usually considered, notably in LDA. Thanks to the the underlying principles of optimal transport, WDA is able to capture both global (at distribution scale) and local (at samples scale) interactions between classes. Regularized Wasserstein distances can be computed using the Sinkhorn matrix scaling algorithm; We show that the optimization of WDA can be tackled using automatic differentiation of Sinkhorn iterations. Numerical experiments show promising results both in terms of prediction and visualization on toy examples and real life datasets such as MNIST and on deep features obtained from a subset of the Caltech dataset.