# Computing the Distance between unbalanced Distributions -- The flat Metric

arXiv: https://arxiv.org/abs/2308.01039 · HTML: https://mlsanity.com/paper/2308.01039/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 37 references that are in the map.
- First posted on arXiv 2 Aug 2023.

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Sorting out Lipschitz function approximation](https://mlsanity.com/paper/1811.05381/) (2018)
- [Lipschitz-Margin Training: Scalable Certification of Perturbation Invariance for Deep Neural Networks](https://mlsanity.com/paper/1802.04034/) (2018)
- [Unbalanced minibatch Optimal Transport; applications to Domain Adaptation](https://mlsanity.com/paper/2103.03606/) (2021)
- [Robust Optimal Transport with Applications in Generative Modeling and Domain Adaptation](https://mlsanity.com/paper/2010.05862/) (2020)
- [Outlier-Robust Optimal Transport](https://mlsanity.com/paper/2012.07363/) (2020)
- [Scalable Unbalanced Optimal Transport using Generative Adversarial Networks](https://mlsanity.com/paper/1810.11447/) (2018)
- [Norm-preserving Orthogonal Permutation Linear Unit Activation Functions (OPLU)](https://mlsanity.com/paper/1604.02313/) (2016)
- [DOTmark - A Benchmark for Discrete Optimal Transport](https://mlsanity.com/paper/1610.03368/) (2016)
- [Imaging with Kantorovich-Rubinstein discrepancy](https://mlsanity.com/paper/1407.0221/) (2014)
- [Unbalanced CO-Optimal Transport](https://mlsanity.com/paper/2205.14923/) (2022)
- [Dual Regularized Optimal Transport](https://mlsanity.com/paper/2012.03126/) (2020)
- [Unbalanced Optimal Transport and Maximum Mean Discrepancies: Interconnections and Rapid Evaluation](https://mlsanity.com/paper/2306.13618/) (2023)

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

- [Guidance Not Obstruction: A Conjugate Consistent Enhanced Strategy for Domain Generalization](https://mlsanity.com/paper/2412.10089/) (2024)

## Abstract (arXiv)

We provide an implementation to compute the flat metric in any dimension. The flat metric, also called dual bounded Lipschitz distance, generalizes the well-known Wasserstein distance W1 to the case that the distributions are of unequal total mass. Thus, our implementation adapts very well to mass differences and uses them to distinguish between different distributions. This is of particular interest for unbalanced optimal transport tasks and for the analysis of data distributions where the sample size is important or normalization is not possible. The core of the method is based on a neural network to determine an optimal test function realizing the distance between two given measures. Special focus was put on achieving comparability of pairwise computed distances from independently trained networks. We tested the quality of the output in several experiments where ground truth was available as well as with simulated data.