# Heterogeneous Wasserstein Discrepancy for Incomparable Distributions

arXiv: https://arxiv.org/abs/2106.02542 · HTML: https://mlsanity.com/paper/2106.02542/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 43 references that are in the map.
- First posted on arXiv 4 Jun 2021.

## Builds on (most cited first, 16 of 16)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [Learning with a Wasserstein Loss](https://mlsanity.com/paper/1506.05439/) (2015)
- [Improving GANs Using Optimal Transport](https://mlsanity.com/paper/1803.05573/) (2018)
- [Gromov-Wasserstein Alignment of Word Embedding Spaces](https://mlsanity.com/paper/1809.00013/) (2018)
- [Max-Sliced Wasserstein Distance and its use for GANs](https://mlsanity.com/paper/1904.05877/) (2019)
- [Scalable Gromov-Wasserstein Learning for Graph Partitioning and Matching](https://mlsanity.com/paper/1905.07645/) (2019)
- [Fused Gromov-Wasserstein distance for structured objects: theoretical foundations and mathematical properties](https://mlsanity.com/paper/1811.02834/) (2018)
- [Subspace Robust Wasserstein Distances](https://mlsanity.com/paper/1901.08949/) (2019)
- [Distributional Sliced-Wasserstein and Applications to Generative Modeling](https://mlsanity.com/paper/2002.07367/) (2020)
- [Sliced Gromov-Wasserstein](https://mlsanity.com/paper/1905.10124/) (2019)
- [Sliced Wasserstein Generative Models](https://mlsanity.com/paper/1904.05408/) (2019)
- [Learning Generative Models across Incomparable Spaces](https://mlsanity.com/paper/1905.05461/) (2019)
- [On Projection Robust Optimal Transport: Sample Complexity and Model Misspecification](https://mlsanity.com/paper/2006.12301/) (2020)
- [Orthogonal Estimation of Wasserstein Distances](https://mlsanity.com/paper/1903.03784/) (2019)

## Built on by (most cited first, 1 of 1)

- [Leveraging Optimal Transport via Projections on Subspaces for Machine Learning Applications](https://mlsanity.com/paper/2311.13883/) (2023)

## Abstract (arXiv)

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 textitsame 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 textitdistributional slicing, embeddings, and on computing the closed-form Wassertein distance between the sliced distributions. We provide a theoretical analysis of this new divergence, called textitheterogeneous 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.