Complete Invariants of Atomic Clouds under Rigid Motion with Lipschitz Continuous Metrics in a Polynomial Time
Abstract
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 Rn. 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.
Fields of Study & Subjects
Advanced Statistical Methods and Models
Soil Geostatistics and Mapping
Statistical and numerical algorithms
Publication details
- Journal / source
- match Communications in Mathematical and in Computer Chemistry
- Publisher
- University Library in Kragujevac
- Volume
- 97
- Issue
- 2
- Pages
- 379–379
- Publication date
- 28 May 2026
- Language
- EN