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

09/03/2020
by   Harish Doraiswamy, et al.
49

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.

READ FULL TEXT

Authors

page 1

page 7

11/02/2021

UnProjection: Leveraging Inverse-Projections for Visual Analytics of High-Dimensional Data

Projection techniques are often used to visualize high-dimensional data,...
03/14/2018

Optimal Bounds for Johnson-Lindenstrauss Transformations

In 1984, Johnson and Lindenstrauss proved that any finite set of data in...
07/16/2021

Measuring and Explaining the Inter-Cluster Reliability of Multidimensional Projections

We propose Steadiness and Cohesiveness, two novel metrics to measure the...
09/02/2021

Knot invariants and their relations: a topological perspective

This work brings methods from topological data analysis to knot theory a...
09/23/2020

Burning sage: Reversing the curse of dimensionality in the visualization of high-dimensional data

In high-dimensional data analysis the curse of dimensionality reasons th...
08/24/2010

Nonlinear Quality of Life Index

We present details of the analysis of the nonlinear quality of life inde...
06/03/2021

Statistical embedding: Beyond principal components

There has been an intense recent activity in embedding of very high dime...

Code Repositories

TopoMap

0-dimensional Homology Preserving Dimensionality Reduction


view repo
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.