# Polynomial-time algorithms for continuous metrics on atomic clouds of unordered points

arXiv: https://arxiv.org/abs/2207.08502 · HTML: https://mlsanity.com/paper/2207.08502/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Jul 2022.

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

- [Is Distance Matrix Enough for Geometric Deep Learning?](https://mlsanity.com/paper/2302.05743/) (2023)
- [On the Completeness of Invariant Geometric Deep Learning Models](https://mlsanity.com/paper/2402.04836/) (2024)
- [Complete Neural Networks for Complete Euclidean Graphs](https://mlsanity.com/paper/2301.13821/) (2023)
- [Completeness of Atomic Structure Representations](https://mlsanity.com/paper/2302.14770/) (2023)
- [On the Expressive Power of Sparse Geometric MPNNs](https://mlsanity.com/paper/2407.02025/) (2024)
- [Quotient Geometry and Persistence-Stable Metrics for Swarm Configurations](https://mlsanity.com/paper/2603.18041/) (2026)
- [Using large language models to probe the limits of atom-centered structural descriptors](https://mlsanity.com/paper/2607.26984/) (2026)
- [LexiCan: Lexicographic Canonicalization of Point Clouds for Machine Learning Applications](https://mlsanity.com/paper/2610.15171/) (2026)

## Abstract (arXiv)

The most fundamental model of a molecule is a cloud of unordered atoms, even without chemical bonds that can depend on thresholds for distances and angles. The strongest equivalence between clouds of atoms is rigid motion, which is a composition of translations and rotations. The existing datasets of experimental and simulated molecules require a continuous quantification of similarity in terms of a distance metric. While clouds of m ordered points were continuously classified by Lagrange's quadratic forms (distance matrices or Gram matrices), their extensions to m unordered points are impractical due to the exponential number of m! permutations. We propose new metrics that are continuous in general position and are computable in a polynomial time in the number m of unordered points in any Euclidean space of a fixed dimension n.