论文标题
康利指数的拓扑推断
Topological Inference of the Conley Index
论文作者
论文摘要
一个孤立的不变集的Conley索引是动态系统研究中的基本对象。在这里，我们考虑在欧几里得空间的闭合子延伸物上的平滑功能，并描述一个框架，用于推断任何紧凑的，连接的，连接的隔离临界集合的conley索引，并从足够大的有限点样本中具有很高的信心。本文的主要结构是特定的索引对，它是有关关键集的本地索引对。我们确定这些索引对具有积极影响，因此接受了强大同源性推断的抽样理论。这使我们能够估算Conley指数，并且是直接的结果，我们还能够使用有限的许多局部评估来估算Morse函数的任何关键点的Morse索引。
The Conley index of an isolated invariant set is a fundamental object in the study of dynamical systems. Here we consider smooth functions on closed submanifolds of Euclidean space and describe a framework for inferring the Conley index of any compact, connected isolated critical set of such a function with high confidence from a sufficiently large finite point sample. The main construction of this paper is a specific index pair which is local to the critical set in question. We establish that these index pairs have positive reach and hence admit a sampling theory for robust homology inference. This allows us to estimate the Conley index, and as a direct consequence, we are also able to estimate the Morse index of any critical point of a Morse function using finitely many local evaluations.
