The most fundamental model of a molecule is a cloud of unordered atoms, even without chemical bonds that can depend on thresholds for distances and angles. The strongest equivalence between clouds of atoms is rigid motion, which is a composition of translations and rotations. The existing datasets of experimental and simulated molecules require a continuous quantification of similarity in terms of a distance metric. While clouds of m ordered points were continuously classified by Lagrange's quadratic forms (distance matrices or Gram matrices), their extensions to m unordered points are impractical due to the exponential number of m! permutations. We propose new metrics that are continuous in general position and are computable in a polynomial time in the number m of unordered points in any Euclidean space of a fixed dimension n.

翻译：分子的最基本模型是无序原子云，即便不依赖取决于距离和角度阈值的化学键也能成立。原子云之间的最强等价关系是刚体运动，即平移与旋转的复合。当前实验与模拟分子数据集需要借助距离度量对相似性进行连续量化。虽然有序点云通过拉格朗日二次型（距离矩阵或格拉姆矩阵）实现连续分类，但其向无序点云的扩展因需处理m!种排列而不可行。我们提出新度量，在一般位置上保持连续性，并且对于固定维数n的欧氏空间中的m个无序点，其计算复杂度为多项式时间。