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arXiv · 2303.13486

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

Vitaliy Kurlin
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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 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.

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Vitaliy Kurlin. 2023-03-23. Complete invariants of atomic clouds under rigid motion with Lipschitz continuous metrics in a polynomial time. https://arxiv.org/abs/2303.13486

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