In this study, we introduce novel methodologies designed to adapt original data in response to the dynamics of persistence diagrams along Wasserstein gradient flows. Our research focuses on the development of algorithms that translate variations in persistence diagrams back into the data space. This advancement enables direct manipulation of the data, guided by observed changes in persistence diagrams, offering a powerful tool for data analysis and interpretation in the context of topological data analysis.

翻译：本研究提出了创新方法，旨在根据沿Wasserstein梯度流的持续性图动态来调整原始数据。我们的研究重点在于开发将持续性图的变化映射回数据空间的算法。这一进展使得能够根据观察到的持续性图变化直接操纵数据，为拓扑数据分析领域的数据分析与解释提供了强大工具。