Kernel Wasserstein Distance

Abstract

The Wasserstein distance is a powerful metric based on the theory of optimal transport. It gives a natural measure of the distance between two distributions with a wide range of applications. In contrast to a number of the common divergences on distributions such as Kullback-Leibler or Jensen-Shannon, it is (weakly) continuous, and thus ideal for analyzing corrupted data. To date, however, no kernel methods for dealing with nonlinear data have been proposed via the Wasserstein distance. In this work, we develop a novel method to compute the L2-Wasserstein distance in a kernel space implemented using the kernel trick. The latter is a general method in machine learning employed to handle data in a nonlinear manner. We evaluate the proposed approach in identifying computerized tomography (CT) slices with dental artifacts in head and neck cancer, performing unsupervised hierarchical clustering on the resulting Wasserstein distance matrix that is computed on imaging texture features extracted from each CT slice. Our experiments show that the kernel approach outperforms classical non-kernel approaches in identifying CT slices with artifacts.