Circular Cohomology Branching and Circular Features
in High Dimensional Data

Bei Wang
SCI Institute, University of Utah
Brian Summa
SCI Institute, University of Utah
Valerio Pascucci
SCI Institute, University of Utah
Mikael Vejdemo-Johansson
Stanford University

Large observations and simulations in scientific research give rise to high-dimensional data sets that present many challenges and opportunities in data analysis and visualization. Researchers in application domains such as engineering, computational biology, climate study, imaging and motion capture are faced with the problem of how to discover compact representations of high-dimensional data while preserving their intrinsic structure. In many applications, the original data is projected onto low-dimensional space via dimensionality reduction techniques prior to modeling. One problem with this approach is that the projection step in the process can fail to preserve structure in the data that is only apparent in high dimensions. Conversely, such techniques may create structural illusions in the projection, implying structure not present in the original high-dimensional data. Our solution is to utilize topological techniques to recover important structures in high-dimensional data that contains non-trivial topology. Specifically, we are interested in high-dimensional branching structures. We construct local circle-valued coordinate functions to represent such features. Subsequently, we perform dimensionality reduction on the data while ensuring such structures are visually preserved. Additionally, we study the effects of global circular structures on visualizations. Our results reveal never-before-seen structures on real-world data sets from a variety of applications.


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author={Bei Wang and Brian Summa and Valerio Pascucci and Mikael Vejdemo-Johansson},
journal={Visualization and Computer Graphics, IEEE Transactions on},
title={Branching and Circular Features in High Dimensional Data},
keywords={circular features;computational biology;data analysis;data visualization;dimensionality reduction techniques;global circular structures;high dimensional data sets;high-dimensional branching structures;high-dimensional data representation;local circle-valued coordinate functions;motion capture;nontrivial topology;topological techniques;computational geometry;data analysis;data structures;data visualisation;topology;},