Heterogeneous Wasserstein Discrepancy for Incomparable Distributions

Mokhtar Z. Alaya, Gilles Gasso, Maxime Berar, Alain Rakotomamonjy

Keywords: Optimal transport, Wasserstein distance, Incomprable distributions, Generative models

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.

One-sentence Summary: A novel optimal transport discrepancy, called heterogenous Wasserstein discrepancy, for comparing probability distributions lying in different spaces.