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.

中文翻译：

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TopoMap：高维数据的 0 维同源性保持投影

多维投影是高维数据分析和可视化的基础工具。除了极少数例外，投影技术旨在将数据从高维空间映射到视觉空间，以保留一些不相似性（相似性）度量，例如欧几里得距离。事实上，尽管采用不同的数学公式来支持数据的不同方面，但大多数多维投影方法都力求保留封装几何属性（例如距离或数据对象之间的邻近关系）的不同度量。然而，几何关系并不是投影中唯一要保留的有趣属性。例如，如果映射过程对拓扑不变量（例如连接组件和循环）提供了一些保证，则可以更可靠地执行对特定结构（例如集群和异常值）的分析。本文介绍了 TopoMap，这是一种新颖的投影技术，可在映射过程中提供拓扑保证。特别地，所提出的方法执行从高维空间到视觉空间的映射，同时保留了高维数据的 Rips 过滤的 0 维持久图，确保过滤在应用时生成相同的连通分量原始数据和投影数据。