Complete invariants of atomic clouds under rigid motion with Lipschitz continuous metrics in a polynomial time
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866917386642259968 |
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| author | Kurlin, Vitaliy |
| author_facet | Kurlin, Vitaliy |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_13486 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Complete invariants of atomic clouds under rigid motion with Lipschitz continuous metrics in a polynomial time Kurlin, Vitaliy Metric Geometry Computational Geometry 51N20, 51K05, 51F20, 51F30, 68U05 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. |
| title | Complete invariants of atomic clouds under rigid motion with Lipschitz continuous metrics in a polynomial time |
| topic | Metric Geometry Computational Geometry 51N20, 51K05, 51F20, 51F30, 68U05 |
| url | https://arxiv.org/abs/2303.13486 |