Title: Quantifying the noise sensitivity of the Wasserstein metric for images
Primary area: other topics in machine learning (i.e., none of the above)
Keywords: optimal transport, earth mover's distance, cryo-electron microscopy, image similarity, robustness

Abstract: Wasserstein metrics are increasingly being used in domains like generative modeling and computer vision as similarity scores for images represented as discrete measures on a grid, yet their behavior under noise remains poorly understood. In this work, we consider the sensitivity of the (signed) Wasserstein distance with respect to pixel-wise additive noise and derive exact (non-asymptotic) bounds. Among other results, we prove that the error in the signed 2-Wasserstein distance scales with the square root of the noise standard deviation, whereas the $L_2$ norm scales linearly. 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 preserve the geometric structure even when the $L_2$ metric fails to do so.

TL;DR: We study the impact of pixel-wise noise when comparing images via the (signed) Wasserstein distance