AI 中文总结
本文提出基于Vaisman阿蒂亚-莫利诺框架的新方法，利用Vaisman重心解决逆问题模糊性，为拓扑数据分析提供新几何范式，可鲁棒恢复隐藏结构并助力高维噪声数据分析。
AI 中文摘要
重构问题是数学与科学的核心，其根本挑战在于：如何从不完整、碎片化或失真的数据中重构隐藏结构？本文提出一种利用Vaisman阿蒂亚-莫利诺框架见解的新方法。与依赖持续同调的传统方法不同，该方法利用Vaisman重心这一内在不变量，其封装了数据集的平均几何特征，以解决逆问题固有的模糊性。本文聚焦于Vaisman重心的理论与应用，为拓扑数据分析提供了一种新视角，摒弃了持续同调，转而采用统一的几何范式。后续论文将把这些思想扩展为通过阿蒂亚-莫利诺框架实现的完整重构方案。该方法为隐藏结构的恢复提供了一种鲁棒且计算易处理的框架，同时为数学科学领域高维、噪声数据的分析开辟了新途径。
英文摘要
Reconstruction problems lie at the very heart of both mathematics and science, posing the fundamental challenge: \emph{How does one reconstruct a hidden structure from incomplete, fragmented, or distorted data?} In this paper, we introduce a new approach that harnesses the insights of the Vaisman Atiyah--Molino framework. In contrast to conventional methods that depend on persistent homology, our approach exploits the concept of the Vaisman centroid---an intrinsic invariant that encapsulates the averaged geometry of a data set---to resolve the inherent ambiguities of inverse problems. In the present paper, we focus on the theory and applications of the Vaisman centroid, offering a new perspective for Topological Data Analysis that eschews persistent homology in favour of a unified geometric paradigm. A subsequent paper will extend these ideas to a complete reconstruction scheme via the Atiyah--Molino framework. Our method provides a robust and computationally tractable framework for the recovery of hidden structures while opening new avenues for the analysis of high-dimensional and noisy data across the mathematical sciences.
DOI：10.1007/978-3-032-03918-7_27
