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.

翻译：任何真实分子的基本表示是无序原子的有限云，其中许多原子在化学上不可区分。度量空间中点云的一种自然等价关系由等距变换（保持距离的变换）定义。在欧几里得空间中，任何等距变换都是平移、旋转和反射的复合。若点是有序的，则该云的等距类由所有成对距离构成的矩阵唯一确定。若m个点是无序的，基于距离矩阵的朴素度量需要指数级数量的m!种排列。我们定义了n维空间中m个无序点云在刚性运动下的完全不变量，该不变量能区分R^n中的所有镜像。关键挑战在于设计一种关于不变量值的距离，使其在噪声下具有Lipschitz连续性，且对于固定维度n，能在云尺寸的多项式时间内计算。