TopoMap: A 0dimensional Homology Preserving Projection of HighDimensional Data
Abstract
Multidimensional Projection is a fundamental tool for highdimensional data analytics and visualization. With very few exceptions, projection techniques are designed to map data from a highdimensional 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 highdimensional space to a visual space, while preserving the 0dimensional persistence diagram of the Rips filtration of the highdimensional 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2020
 DOI:
 10.48550/arXiv.2009.01512
 arXiv:
 arXiv:2009.01512
 Bibcode:
 2020arXiv200901512D
 Keywords:

 Computer Science  Graphics;
 Computer Science  Computational Geometry;
 Computer Science  Computer Vision and Pattern Recognition;
 Computer Science  HumanComputer Interaction;
 Computer Science  Machine Learning