# On the 1-Wasserstein Distance between Location-Scale Distributions and the Effect of Differential Privacy

arXiv: https://arxiv.org/abs/2304.14869 · HTML: https://mlsanity.com/paper/2304.14869/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 26 references that are in the map.
- First posted on arXiv 28 Apr 2023.

## Builds on (most cited first, 5 of 5)

- [Statistical Aspects of Wasserstein Distances](https://mlsanity.com/paper/1806.05500/) (2018)
- [On the folded normal distribution](https://mlsanity.com/paper/1402.3559/) (2014)
- [Deconstructing Generative Adversarial Networks](https://mlsanity.com/paper/1901.09465/) (2019)
- [Distributionally-Robust Machine Learning Using Locally Differentially-Private Data](https://mlsanity.com/paper/2006.13488/) (2020)
- [Market Value of Differentially-Private Smart Meter Data](https://mlsanity.com/paper/2104.09898/) (2021)

## Built on by (most cited first, 7 of 7)

- [Disentangling impact of capacity, objective, batchsize, estimators, and step-size on flow VI](https://mlsanity.com/paper/2412.08824/) (2024)
- [Towards a framework on tabular synthetic data generation: a minimalist approach: theory, use cases, and limitations](https://mlsanity.com/paper/2411.10982/) (2024)
- [A Robust Quantile Huber Loss With Interpretable Parameter Adjustment In Distributional Reinforcement Learning](https://mlsanity.com/paper/2401.02325/) (2024)
- [Deterministic Uncertainty Propagation for Improved Model-Based Offline Reinforcement Learning](https://mlsanity.com/paper/2406.04088/) (2024)
- [Bayesian Experimental Design for Model Discrepancy Calibration: An Auto-Differentiable Ensemble Kalman Inversion Approach](https://mlsanity.com/paper/2504.20319/) (2025)
- [Uncertainty-driven Embedding Convolution](https://mlsanity.com/paper/2507.20718/) (2025)
- [Bayesian Experimental Design for Model Discrepancy Calibration: A Rivalry between Kullback--Leibler Divergence and Wasserstein Distance](https://mlsanity.com/paper/2601.16425/) (2026)

## Abstract (arXiv)

We provide an exact expressions for the 1-Wasserstein distance between independent location-scale distributions. The expressions are represented using location and scale parameters and special functions such as the standard Gaussian CDF or the Gamma function. Specifically, we find that the 1-Wasserstein distance between independent univariate location-scale distributions is equivalent to the mean of a folded distribution within the same family whose underlying location and scale are equal to the difference of the locations and scales of the original distributions. A new linear upper bound on the 1-Wasserstein distance is presented and the asymptotic bounds of the 1-Wasserstein distance are detailed in the Gaussian case. The effect of differential privacy using the Laplace and Gaussian mechanisms on the 1-Wasserstein distance is studied using the closed-form expressions and bounds.