title: Heterogeneous Wasserstein Discrepancy for Incomparable Distributions
abstract: Optimal Transport (OT) metrics allow for defining discrepancies between two probability measures. Wasserstein distance is for longer the celebrated OT-distance frequently-used in the literature, which seeks probability distributions to be supported on the $\text{\it same}$ metric space. Because of its high computational complexity, several approximate Wasserstein distances have been proposed based on entropy regularization or on slicing, and one-dimensional Wassserstein computation. In this paper, we propose a novel extension of Wasserstein distance to compare two incomparable distributions, that hinges on the idea of $\text{\it distributional slicing}$, embeddings, and on computing the closed-form Wassertein distance between the sliced distributions.  We provide a theoretical analysis of this new divergence, called $\text{\it heterogeneous Wasserstein discrepancy (HWD)}$, and we show that it preserves several interesting properties including rotation-invariance.  We show that  the embeddings involved in HWD can be  efficiently learned.  Finally, we provide a large set of experiments illustrating the behavior of HWD as a divergence in the context of generative modeling and in query framework.
authors: Mokhtar Z. Alaya Gilles Gasso Maxime Berar Alain Rakotomamonjy
hIndex: Alain Rakotomamonjy 32