# On the stability of scale-space metrics

arXiv: https://arxiv.org/abs/2606.27605 · HTML: https://mlsanity.com/paper/2606.27605/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 52 references that are in the map.
- First posted on arXiv 25 Jun 2026.

## Builds on (most cited first, 11 of 11)

- [Deep Scattering Spectrum](https://mlsanity.com/paper/1304.6763/) (2013)
- [Rotationally Invariant Image Representation for Viewing Direction Classification in Cryo-EM](https://mlsanity.com/paper/1309.7643/) (2013)
- [Hierarchical Coupled Geometry Analysis for Neuronal Structure and Activity Pattern Discovery](https://mlsanity.com/paper/1511.02086/) (2015)
- [Earthmover-based manifold learning for analyzing molecular conformation spaces](https://mlsanity.com/paper/1911.06107/) (2019)
- [Alignment of Density Maps in Wasserstein Distance](https://mlsanity.com/paper/2305.12310/) (2023)
- [Wasserstein K-Means for Clustering Tomographic Projections](https://mlsanity.com/paper/2010.09989/) (2020)
- [Data-Driven Tree Transforms and Metrics](https://mlsanity.com/paper/1708.05768/) (2017)
- [Fast rigid alignment of heterogeneous images in sliced Wasserstein distance](https://mlsanity.com/paper/2503.13756/) (2025)
- [Mahalanobis Distance for Class Averaging of Cryo-EM Images](https://mlsanity.com/paper/1611.03193/) (2016)
- [Joint Hierarchical Representation Learning of Samples and Features via Informed Tree-Wasserstein Distance](https://mlsanity.com/paper/2501.03627/) (2025)
- [Cramer Type Distances for Learning Gaussian Mixture Models by Gradient Descent](https://mlsanity.com/paper/2307.06753/) (2023)

## Abstract (arXiv)

We study the stability of a classical family of metrics defined over functions' Gaussian scale-space representations, focusing on the comparison of images (functions of two variables). These metrics have precedents both in harmonic analysis, specifically the theory of Besov spaces, and in classical methods of image processing; special cases are also known to be metrically equivalent to certain Wasserstein distances. We quantify these metrics' robustness to geometric deformations, and introduce rotationally-invariant versions that are stable to changes in angle when comparing tomographic projections. We also describe computationally efficient algorithms for evaluating the metrics from finite samples, and prove their robustness to additive noise. The results are illustrated through numerical experiments.