Simplexwise Distance Distributions for finite spaces with metrics and measures

03/24/2023

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by   Vitaliy Kurlin, et al.

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A finite set of unlabelled points in Euclidean space is the simplest representation of many real objects from mineral rocks to sculptures. Since most solid objects are rigid, their natural equivalence is rigid motion or isometry maintaining all inter-point distances. More generally, any finite metric space is an example of a metric-measure space that has a probability measure and a metric satisfying all axioms. This paper develops Simplexwise Distance Distributions (SDDs) for any finite metric spaces and metric-measures spaces. These SDDs classify all known non-equivalent spaces that were impossible to distinguish by simpler invariants. We define metrics on SDDs that are Lipschitz continuous and allow exact computations whose parametrised complexities are polynomial in the number of given points.