TopoMap: A 0-dimensional Homology Preserving Projection of High-Dimensional Data - Archive ouverte HAL Access content directly
Journal Articles IEEE Transactions on Visualization and Computer Graphics Year : 2020

TopoMap: A 0-dimensional Homology Preserving Projection of High-Dimensional Data

Harish Doraiswamy
  • Function : Author
Julien Tierny
Paulo J S Silva
  • Function : Author
Gustavo Nonato
  • Function : Author
Claudio Silva
  • Function : Author

Abstract

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.
Fichier principal
Vignette du fichier
2009.01512.pdf (30.12 Mo) Télécharger le fichier
Origin : Files produced by the author(s)
Loading...

Dates and versions

hal-02949185 , version 1 (25-09-2020)

Identifiers

  • HAL Id : hal-02949185 , version 1

Cite

Harish Doraiswamy, Julien Tierny, Paulo J S Silva, Gustavo Nonato, Claudio Silva. TopoMap: A 0-dimensional Homology Preserving Projection of High-Dimensional Data. IEEE Transactions on Visualization and Computer Graphics, 2020. ⟨hal-02949185⟩
69 View
5 Download

Share

Gmail Facebook X LinkedIn More