# Quantifying the noise sensitivity of the Wasserstein metric for images

arXiv: https://arxiv.org/abs/2510.01015 · HTML: https://mlsanity.com/paper/2510.01015/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 43 references that are in the map.
- First posted on arXiv 1 Oct 2025.

## Builds on (most cited first, 21 of 21)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures](https://mlsanity.com/paper/1508.07941/) (2015)
- [Stabilized Sparse Scaling Algorithms for Entropy Regularized Transport Problems](https://mlsanity.com/paper/1610.06519/) (2016)
- [Conditional Wasserstein Distances with Applications in Bayesian OT Flow Matching](https://mlsanity.com/paper/2403.18705/) (2024)
- [DOTmark - A Benchmark for Discrete Optimal Transport](https://mlsanity.com/paper/1610.03368/) (2016)
- [Flow Matching with Semidiscrete Couplings](https://mlsanity.com/paper/2509.25519/) (2025)
- [Fast and Scalable Optimal Transport for Brain Tractograms](https://mlsanity.com/paper/2107.02010/) (2021)
- [A Transportation Lp Distance for Signal Analysis](https://mlsanity.com/paper/1609.08669/) (2016)
- [On Fitting Flow Models with Large Sinkhorn Couplings](https://mlsanity.com/paper/2506.05526/) (2025)
- [Manifold learning with arbitrary norms](https://mlsanity.com/paper/2012.14172/) (2020)
- [Alignment of Density Maps in Wasserstein Distance](https://mlsanity.com/paper/2305.12310/) (2023)
- [AlignOT: An optimal transport based algorithm for fast 3D alignment with applications to cryogenic electron microscopy density maps](https://mlsanity.com/paper/2210.09361/) (2022)
- [A New Robust Partial p-Wasserstein-Based Metric for Comparing Distributions](https://mlsanity.com/paper/2405.03664/) (2024)
- [Wasserstein K-Means for Clustering Tomographic Projections](https://mlsanity.com/paper/2010.09989/) (2020)
- [Fast rigid alignment of heterogeneous images in sliced Wasserstein distance](https://mlsanity.com/paper/2503.13756/) (2025)
- [Sharp Convergence Rates of Empirical Unbalanced Optimal Transport for Spatio-Temporal Point Processes](https://mlsanity.com/paper/2509.04225/) (2025)

## Abstract (arXiv)

Wasserstein metrics are increasingly adopted as similarity scores for images. We consider the sensitivity of Wasserstein metrics with respect to pixel-wise additive noise when the images are treated as discrete measures on the pixel grid. We derive finite-sample expectation bounds for a Gaussian noise model. Among other results, we prove that the error in the signed 2-Wasserstein discrepancy scales with the square root of the noise standard deviation. This is favorable compared to the Euclidean metric that scales linearly, and thus provides a theoretical basis for the benefits of optimal transport distances in noisy settings. We present experiments that support our theoretical findings and point to a peculiar phenomenon where increasing the level of noise can decrease the Wasserstein distance. A case study on cryo-electron microscopy images demonstrates that the Wasserstein metric can capture the geometry of the data manifold in high noise settings even when the Euclidean metric fails.