Topological Inference of the Conley Index

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2023-09-23 DOI:10.1007/s10884-023-10310-1

Ka Man Yim, Vidit Nanda

引用次数: 0

Abstract

Abstract 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.

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康利指数的拓扑推断

孤立不变集的Conley指标是动力系统研究中的一个基本对象。本文考虑欧几里得空间闭子流形上的光滑函数，并描述了从足够大的有限点样本上推断此类函数的任何紧的、连通的、高置信度的孤立临界集的Conley指数的框架。本文的主要构造是一个特定的指标对，它局部于所讨论的临界集。我们证明了这些指标对具有正到达性，并因此承认了鲁棒同调推理的抽样理论。这允许我们估计康利指数，作为一个直接的结果，我们也能够估计莫尔斯函数的任何临界点莫尔斯指数使用有限多个局部计算。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.