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.

翻译：欧氏空间中有限个未标记点是从矿物岩石到雕塑等众多现实物体的最简单表示。由于大多数固体物体具有刚性，其自然等价关系为保持所有点间距离的刚体运动或等距变换。更一般地，任何有限度量空间都是度量-测度空间的一个实例，此类空间既包含概率测度又满足所有公理的度量。本文为任意有限度量空间和度量-测度空间发展了逐单纯形距离分布（Simplexwise Distance Distributions, SDDs）。这些SDD能够分类所有已知的非等价空间，而这些空间此前无法通过更简单的不变量加以区分。我们定义了SDD上的度量，该度量具有Lipschitz连续性，并允许精确计算，其参数化复杂度关于给定点数为多项式阶。