# Normalized Wasserstein Distance for Mixture Distributions with Applications in Adversarial Learning and Domain Adaptation

arXiv: https://arxiv.org/abs/1902.00415 · HTML: https://mlsanity.com/paper/1902.00415/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Feb 2019.

## Built on by (most cited first, 23 of 23)

- [Domain Generalization: A Survey](https://mlsanity.com/paper/2103.02503/) (2021)
- [Aggregating From Multiple Target-Shifted Sources](https://mlsanity.com/paper/2105.04051/) (2021)
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- [Make the U in UDA Matter: Invariant Consistency Learning for Unsupervised Domain Adaptation](https://mlsanity.com/paper/2309.12742/) (2023)
- [SLOSH: Set LOcality Sensitive Hashing via Sliced-Wasserstein Embeddings](https://mlsanity.com/paper/2112.05872/) (2021)
- [Graphical Modeling for Multi-Source Domain Adaptation](https://mlsanity.com/paper/2104.13057/) (2021)
- [Dynamic Instance Domain Adaptation](https://mlsanity.com/paper/2203.05028/) (2022)
- [CODA: Temporal Domain Generalization via Concept Drift Simulator](https://mlsanity.com/paper/2310.01508/) (2023)
- [Domain Generalization with Small Data](https://mlsanity.com/paper/2402.06150/) (2024)
- [Optimal Transport-Guided Source-Free Adaptation for Face Anti-Spoofing](https://mlsanity.com/paper/2503.22984/) (2025)
- [Is There a Better Source Distribution than Gaussian? Exploring Source Distributions for Image Flow Matching](https://mlsanity.com/paper/2512.18184/) (2025)
- [Domain Attention Consistency for Multi-Source Domain Adaptation](https://mlsanity.com/paper/2111.03911/) (2021)
- [Robust Target Training for Multi-Source Domain Adaptation](https://mlsanity.com/paper/2210.01676/) (2022)
- [Fast, accurate and lightweight sequential simulation-based inference using Gaussian locally linear mappings](https://mlsanity.com/paper/2403.07454/) (2024)
- [MetaPix: Domain Transfer for Semantic Segmentation by Meta Pixel Weighting](https://mlsanity.com/paper/2110.01777/) (2021)
- [Pruning for Better Domain Generalizability](https://mlsanity.com/paper/2306.13237/) (2023)
- [Contrastive Conditional Alignment based on Label Shift Calibration for Imbalanced Domain Adaptation](https://mlsanity.com/paper/2412.20337/) (2024)
- [Geometric Moment Alignment for Domain Adaptation via Siegel Embeddings](https://mlsanity.com/paper/2510.14666/) (2025)
- [Generative Model Based Noise Robust Training for Unsupervised Domain Adaptation](https://mlsanity.com/paper/2303.05734/) (2023)
- [Graceful Degradation and Related Fields](https://mlsanity.com/paper/2106.11119/) (2021)
- [Graph-Informed Adversarial Modeling: Infimal Subadditivity of Interpolative Divergences](https://mlsanity.com/paper/2603.20025/) (2026)

## Abstract (arXiv)

Understanding proper distance measures between distributions is at the core of several learning tasks such as generative models, domain adaptation, clustering, etc. In this work, we focus on mixture distributions that arise naturally in several application domains where the data contains different sub-populations. For mixture distributions, established distance measures such as the Wasserstein distance do not take into account imbalanced mixture proportions. Thus, even if two mixture distributions have identical mixture components but different mixture proportions, the Wasserstein distance between them will be large. This often leads to undesired results in distance-based learning methods for mixture distributions. In this paper, we resolve this issue by introducing the Normalized Wasserstein measure. The key idea is to introduce mixture proportions as optimization variables, effectively normalizing mixture proportions in the Wasserstein formulation. Using the proposed normalized Wasserstein measure leads to significant performance gains for mixture distributions with imbalanced mixture proportions compared to the vanilla Wasserstein distance. We demonstrate the effectiveness of the proposed measure in GANs, domain adaptation and adversarial clustering in several benchmark datasets.