Heterogeneous Wasserstein Discrepancy for Incomparable Distributions
This work addresses a limitation in optimal transport metrics for researchers and practitioners dealing with incomparable distributions, offering a new tool with potential applications in machine learning domains.
The paper tackles the problem of comparing probability distributions that are not supported on the same metric space by proposing a novel extension of Wasserstein distance, called heterogeneous Wasserstein discrepancy (HWD), which uses distributional slicing and embeddings to compute distances efficiently. The result includes theoretical properties like rotation-invariance and experimental validation in generative modeling and query frameworks.
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 $\textit{same}$ 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 $\textit{distributional slicing}$, embeddings, and on computing the closed-form Wassertein distance between the sliced distributions. We provide a theoretical analysis of this new divergence, called $\textit{heterogeneous Wasserstein discrepancy (HWD)}$, 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.