# Alignment of Density Maps in Wasserstein Distance

arXiv: https://arxiv.org/abs/2305.12310 · HTML: https://mlsanity.com/paper/2305.12310/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 42 references that are in the map.
- First posted on arXiv 21 May 2023.

## Builds on (most cited first, 13 of 13)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [A Tutorial on Bayesian Optimization](https://mlsanity.com/paper/1807.02811/) (2018)
- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [Manopt, a Matlab toolbox for optimization on manifolds](https://mlsanity.com/paper/1308.5200/) (2013)
- [Pymanopt: A Python Toolbox for Optimization on Manifolds using Automatic Differentiation](https://mlsanity.com/paper/1603.03236/) (2016)
- [Geometry-aware Bayesian Optimization in Robotics using Riemannian Mat'ern Kernels](https://mlsanity.com/paper/2111.01460/) (2021)
- [Wassmap: Wasserstein Isometric Mapping for Image Manifold Learning](https://mlsanity.com/paper/2204.06645/) (2022)
- [Manifold learning with arbitrary norms](https://mlsanity.com/paper/2012.14172/) (2020)
- [Earthmover-based manifold learning for analyzing molecular conformation spaces](https://mlsanity.com/paper/1911.06107/) (2019)
- [Centering noisy images with application to cryo-EM](https://mlsanity.com/paper/2009.04810/) (2020)
- [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)
- [Wasserstein K-Means for Clustering Tomographic Projections](https://mlsanity.com/paper/2010.09989/) (2020)
- [Three-Dimensional Alignment of Density Maps in Cryo-Electron Microscopy](https://mlsanity.com/paper/2206.08027/) (2022)

## Built on by (most cited first, 8 of 8)

- [Enhancing Gaussian Process Surrogates for Optimization and Posterior Approximation via Random Exploration](https://mlsanity.com/paper/2401.17037/) (2024)
- [Fast rigid alignment of heterogeneous images in sliced Wasserstein distance](https://mlsanity.com/paper/2503.13756/) (2025)
- [Robust Point Matching with Distance Profiles](https://mlsanity.com/paper/2312.12641/) (2023)
- [Quantization-based Bounds on the Wasserstein Metric](https://mlsanity.com/paper/2506.00976/) (2025)
- [Quantifying the noise sensitivity of the Wasserstein metric for images](https://mlsanity.com/paper/2510.01015/) (2025)
- [The Joint Gromov Wasserstein Objective for Multiple Object Matching](https://mlsanity.com/paper/2511.16868/) (2025)
- [On the stability of scale-space metrics](https://mlsanity.com/paper/2606.27605/) (2026)
- [Cluster-Aware Matching via Laplacian Optimal Transport](https://mlsanity.com/paper/2607.16178/) (2026)

## Abstract (arXiv)

In this paper we propose an algorithm for aligning three-dimensional objects when represented as density maps, motivated by applications in cryogenic electron microscopy. The algorithm is based on minimizing the 1-Wasserstein distance between the density maps after a rigid transformation. The induced loss function enjoys a more benign landscape than its Euclidean counterpart and Bayesian optimization is employed for computation. Numerical experiments show improved accuracy and efficiency over existing algorithms on the alignment of real protein molecules. In the context of aligning heterogeneous pairs, we illustrate a potential need for new distance functions.