# Relative Translation Invariant Wasserstein Distance

arXiv: https://arxiv.org/abs/2409.02416 · HTML: https://mlsanity.com/paper/2409.02416/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 65 references that are in the map.
- First posted on arXiv 4 Sep 2024.

## Builds on (most cited first, 30 of 30)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [SuperGlue: Learning Feature Matching with Graph Neural Networks](https://mlsanity.com/paper/1911.11763/) (2019)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [Wasserstein Auto-Encoders](https://mlsanity.com/paper/1711.01558/) (2017)
- [Joint Distribution Optimal Transportation for Domain Adaptation](https://mlsanity.com/paper/1705.08848/) (2017)
- [Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration](https://mlsanity.com/paper/1705.09634/) (2017)
- [DeepJDOT: Deep Joint Distribution Optimal Transport for Unsupervised Domain Adaptation](https://mlsanity.com/paper/1803.10081/) (2018)
- [Distributionally Robust Logistic Regression](https://mlsanity.com/paper/1509.09259/) (2015)
- [Gromov-Wasserstein Learning for Graph Matching and Node Embedding](https://mlsanity.com/paper/1901.06003/) (2019)
- [Unsupervised Alignment of Embeddings with Wasserstein Procrustes](https://mlsanity.com/paper/1805.11222/) (2018)
- [Robust Optimal Transport with Applications in Generative Modeling and Domain Adaptation](https://mlsanity.com/paper/2010.05862/) (2020)
- [Large-Scale Wasserstein Gradient Flows](https://mlsanity.com/paper/2106.00736/) (2021)
- [Unsupervised Domain Adaptation with Similarity Learning](https://mlsanity.com/paper/1711.08995/) (2017)
- [Sliced Gromov-Wasserstein](https://mlsanity.com/paper/1905.10124/) (2019)
- [The Unbalanced Gromov Wasserstein Distance: Conic Formulation and Relaxation](https://mlsanity.com/paper/2009.04266/) (2020)
- [Variational Wasserstein gradient flow](https://mlsanity.com/paper/2112.02424/) (2021)
- [Continuous Wasserstein-2 Barycenter Estimation without Minimax Optimization](https://mlsanity.com/paper/2102.01752/) (2021)
- [Outlier-Robust Optimal Transport](https://mlsanity.com/paper/2012.07363/) (2020)
- [Energy-Based Sliced Wasserstein Distance](https://mlsanity.com/paper/2304.13586/) (2023)
- [Towards Optimal Transport with Global Invariances](https://mlsanity.com/paper/1806.09277/) (2018)
- [Fixed-Support Wasserstein Barycenters: Computational Hardness and Fast Algorithm](https://mlsanity.com/paper/2002.04783/) (2020)
- [Revisiting Sliced Wasserstein on Images: From Vectorization to Convolution](https://mlsanity.com/paper/2204.01188/) (2022)
- [Wasserstein Iterative Networks for Barycenter Estimation](https://mlsanity.com/paper/2201.12245/) (2022)
- [Spatio-Temporal Alignments: Optimal transport through space and time](https://mlsanity.com/paper/1910.03860/) (2019)
- [First-Order Methods for Wasserstein Distributionally Robust MDP](https://mlsanity.com/paper/2009.06790/) (2020)
- [On Unbalanced Optimal Transport: Gradient Methods, Sparsity and Approximation Error](https://mlsanity.com/paper/2202.03618/) (2022)
- [Entropic Gromov-Wasserstein between Gaussian Distributions](https://mlsanity.com/paper/2108.10961/) (2021)
- [Parallelly Sliced Optimal Transport on Spheres and on the Rotation Group](https://mlsanity.com/paper/2401.16896/) (2024)
- [Neural approximation of Wasserstein distance via a universal architecture for symmetric and factorwise group invariant functions](https://mlsanity.com/paper/2308.00273/) (2023)

## Built on by (most cited first, 3 of 3)

- [Bayesian Experimental Design for Model Discrepancy Calibration: An Auto-Differentiable Ensemble Kalman Inversion Approach](https://mlsanity.com/paper/2504.20319/) (2025)
- [Bayesian Experimental Design for Model Discrepancy Calibration: A Rivalry between Kullback--Leibler Divergence and Wasserstein Distance](https://mlsanity.com/paper/2601.16425/) (2026)
- [Relative Wasserstein Angle and the Problem of the W2-Nearest Gaussian Distribution](https://mlsanity.com/paper/2601.22355/) (2026)

## Abstract (arXiv)

Motivated by the Bures distance, we introduce a new family of distances, relative translation invariant Wasserstein distances, denoted by RWp, as an extension of the classical Wasserstein distances Wp for p ∈ [1, +∞). We establish that RWp defines a valid metric and demonstrate that this type of metric is more intrinsic than the classical Wasserstein distance. A bi-level algorithm is designed to compute the general RWp distance between arbitrary discrete distributions. Moreover, when p = 2, we show that the optimal coupling matrix is invariant under distributional translation in the discrete setting, and we further propose two algorithms, the RW2-LP algorithm and the RW2-Sinkhorn algorithm, to improve the numerical stability of computing W2 distance and the optimal coupling matrix solutions. Finally, we conduct three experiments to validate our theoretical results and algorithms. The first two experiments report that the RW2-LP algorithm and the RW2-Sinkhorn algorithm, both with and without normalization, can significantly reduce the numerical errors compared to standard algorithms. The third experiment shows that RWp algorithms are computationally scalable and applicable to the retrieval of similar thunderstorm patterns in practical applications.