# Complete invariants of atomic clouds under rigid motion with Lipschitz continuous metrics in a polynomial time

arXiv: https://arxiv.org/abs/2303.13486 · HTML: https://mlsanity.com/paper/2303.13486/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Mar 2023.

## Built on by (most cited first, 2 of 2)

- [Recognizing Rigid Patterns of Unlabeled Point Clouds by Complete and Continuous Isometry Invariants with no False Negatives and no False Positives](https://mlsanity.com/paper/2303.15385/) (2023)
- [Using large language models to probe the limits of atom-centered structural descriptors](https://mlsanity.com/paper/2607.26984/) (2026)

## Abstract (arXiv)

A basic representation of any real molecule is a finite cloud of unordered atoms, many of which are chemically indistinguishable. A natural equivalence on point clouds in any metric space is defined by isometries that are distance-preserving transformations. In a Euclidean space, any isometry is a composition of translations, rotations, and reflections. If points are ordered, the isometry class of this cloud is uniquely determined by the matrix of all pairwise distances. If m points are unordered, a naive metric based on distance matrices needs exponentially many m! permutations. We define a complete invariant for n-dimensional clouds of m unordered points under rigid motion, which distinguishes all mirror images in R^n. The key challenge was to design a distance on invariant values that is Lipschitz continuous under noise and computable in a polynomial time of cloud sizes, for a fixed dimension n.