# A New Robust Partial p-Wasserstein-Based Metric for Comparing Distributions

arXiv: https://arxiv.org/abs/2405.03664 · HTML: https://mlsanity.com/paper/2405.03664/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 42 references that are in the map.
- First posted on arXiv 6 May 2024.

## Builds on (most cited first, 22 of 22)

- [DINOv2: Learning Robust Visual Features without Supervision](https://mlsanity.com/paper/2304.07193/) (2023)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations](https://mlsanity.com/paper/1505.05116/) (2015)
- [Fast Computation of Wasserstein Barycenters](https://mlsanity.com/paper/1310.4375/) (2013)
- [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)
- [Differential Properties of Sinkhorn Approximation for Learning with Wasserstein Distance](https://mlsanity.com/paper/1805.11897/) (2018)
- [Hierarchical Optimal Transport for Document Representation](https://mlsanity.com/paper/1906.10827/) (2019)
- [Outlier-Robust Optimal Transport](https://mlsanity.com/paper/2012.07363/) (2020)
- [Robust Contrastive Learning against Noisy Views](https://mlsanity.com/paper/2201.04309/) (2022)
- [Generative Modeling through the Semi-dual Formulation of Unbalanced Optimal Transport](https://mlsanity.com/paper/2305.14777/) (2023)
- [On Robust Optimal Transport: Computational Complexity and Barycenter Computation](https://mlsanity.com/paper/2102.06857/) (2021)
- [Outlier-Robust Optimal Transport: Duality, Structure, and Statistical Analysis](https://mlsanity.com/paper/2111.01361/) (2021)
- [Stochastic Wasserstein Barycenters](https://mlsanity.com/paper/1802.05757/) (2018)
- [Scalable Nearest Neighbor Search for Optimal Transport](https://mlsanity.com/paper/1910.04126/) (2019)
- [Semi-relaxed Gromov-Wasserstein divergence with applications on graphs](https://mlsanity.com/paper/2110.02753/) (2021)
- [Wasserstein regularization for sparse multi-task regression](https://mlsanity.com/paper/1805.07833/) (2018)
- [CSOT: Curriculum and Structure-Aware Optimal Transport for Learning with Noisy Labels](https://mlsanity.com/paper/2312.06221/) (2023)
- [Wasserstein K-means for clustering probability distributions](https://mlsanity.com/paper/2209.06975/) (2022)
- [Outlier-Robust Wasserstein DRO](https://mlsanity.com/paper/2311.05573/) (2023)
- [A Graph Theoretic Additive Approximation of Optimal Transport](https://mlsanity.com/paper/1905.11830/) (2019)
- [DECWA : Density-Based Clustering using Wasserstein Distance](https://mlsanity.com/paper/2310.16552/) (2023)

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

- [Metric properties of partial and robust Gromov-Wasserstein distances](https://mlsanity.com/paper/2411.02198/) (2024)
- [Conic Formulations of Transport Metrics for Unbalanced Measure Networks and Hypernetworks](https://mlsanity.com/paper/2508.10888/) (2025)
- [Cross-Fusion Distance: A Novel Metric for Measuring Fusion and Separability Between Data Groups in Representation Space](https://mlsanity.com/paper/2601.22036/) (2026)
- [Out-of-Distribution Detection with Overlap Index](https://mlsanity.com/paper/2412.06168/) (2024)
- [Quantifying the noise sensitivity of the Wasserstein metric for images](https://mlsanity.com/paper/2510.01015/) (2025)
- [Accelerated Sinkhorn Algorithms for Partial Optimal Transport](https://mlsanity.com/paper/2601.17196/) (2026)
- [Partial Soft-Matching Distance for Neural Representational Comparison with Partial Unit Correspondence](https://mlsanity.com/paper/2602.19331/) (2026)

## Abstract (arXiv)

The 2-Wasserstein distance is sensitive to minor geometric differences between distributions, making it a very powerful dissimilarity metric. However, due to this sensitivity, a small outlier mass can also cause a significant increase in the 2-Wasserstein distance between two similar distributions. Similarly, sampling discrepancy can cause the empirical 2-Wasserstein distance on n samples in ℝ2 to converge to the true distance at a rate of n-1/4, which is significantly slower than the rate of n-1/2 for 1-Wasserstein distance. We introduce a new family of distances parameterized by k ≥ 0, called k-RPW that is based on computing the partial 2-Wasserstein distance. We show that (1) k-RPW satisfies the metric properties, (2) k-RPW is robust to small outlier mass while retaining the sensitivity of 2-Wasserstein distance to minor geometric differences, and (3) when k is a constant, k-RPW distance between empirical distributions on n samples in ℝ2 converges to the true distance at a rate of n-1/3, which is faster than the convergence rate of n-1/4 for the 2-Wasserstein distance. Using the partial p-Wasserstein distance, we extend our distance to any p ∈ [1,∞]. By setting parameters k or p appropriately, we can reduce our distance to the total variation, p-Wasserstein, and the L'evy-Prokhorov distances. Experiments show that our distance function achieves higher accuracy in comparison to the 1-Wasserstein, 2-Wasserstein, and TV distances for image retrieval tasks on noisy real-world data sets.