TopoMap：高维数据的0维保同调投影
TopoMap: A 0-dimensional Homology Preserving Projection of High-Dimensional Data
摘要：
多维投影是高维数据分析和可视化的基本工具。除了极少数例外，投影技术被设计为将数据从高维空间映射到视觉空间，以便保留一些相异性（相似性）度量，例如欧几里得距离。事实上，虽然采用不同的数学公式，旨在有利于数据的不同方面，大多数多维投影方法努力保持封装几何属性，如距离或数据对象之间的邻近关系的相异性措施。然而，几何关系并不是投影中唯一需要保留的有趣属性。例如，如果映射过程对拓扑不变量（如连通分量和回路）提供了某种保证，则可以更可靠地执行对特定结构（如聚类和离群值）的分析。本文介绍了TopoMap，一种新的投影技术，提供拓扑保证在映射过程中。特别地，所提出的方法执行从高维空间到视觉空间的映射，同时保留高维数据的Rips过滤的0维持久性图，确保过滤在应用于原始数据和投影数据时生成相同的连通分量。所提出的案例研究表明，拓扑保证提供的TopoMap不仅带来了信心的视觉分析过程，但也可以用来协助其他投影方法的评估。
英文摘要：
Multidimensional Projection is a fundamental tool for high-dimensional data analytics and visualization. With very few exceptions, projection techniques are designed to map data from a high-dimensional space to a visual space so as to preserve some dissimilarity (similarity) measure, such as the Euclidean distance for example. In fact, although adopting distinct mathematical formulations designed to favor different aspects of the data, most multidimensional projection methods strive to preserve dissimilarity measures that encapsulate geometric properties such as distances or the proximity relation between data objects. However, geometric relations are not the only interesting property to be preserved in a projection. For instance, the analysis of particular structures such as clusters and outliers could be more reliably performed if the mapping process gives some guarantee as to topological invariants such as connected components and loops. This paper introduces TopoMap, a novel projection technique which provides topological guarantees during the mapping process. In particular, the proposed method performs the mapping from a high-dimensional space to a visual space, while preserving the 0-dimensional persistence diagram of the Rips filtration of the high-dimensional data, ensuring that the filtrations generate the same connected components when applied to the original as well as projected data. The presented case studies show that the topological guarantee provided by TopoMap not only brings confidence to the visual analytic process but also can be used to assist in the assessment of other projection methods.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
