原子云在具有Lipschitz连续度量的刚体运动下的完整不变量
Complete invariants of atomic clouds under rigid motion with Lipschitz continuous metrics in a polynomial time
AI总结：
本文研究了在具有Lipschitz连续度量的刚体运动下，原子云的完整不变量问题，提出了一种能够在多项式时间内计算的度量方法，以区分所有镜像。
摘要：
任何真实分子的基本表示都是有限的无序原子云，其中许多原子在化学上是无法区分的。在任何度量空间中，点云的自然等价性由保持距离的等距变换定义。在欧几里得空间中，任何等距变换都是平移、旋转和反射的组合。如果点是有序的，该云的等距类由所有成对距离的矩阵唯一确定。如果m个点是无序的，基于距离矩阵的朴素度量需要指数级的m!次排列。我们定义了n维m个无序点云在刚体运动下的完整不变量，该不变量在R^n中区分所有镜像。关键挑战是设计一种在噪声下Lipschitz连续且在固定维度n下可多项式时间计算的度量。
英文摘要：
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.
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