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Sparse Circular Coordinates via Principal \(\mathbb {Z}\)-Bundles

  • Jose A. PereaEmail author
Conference paper
  • 67 Downloads
Part of the Abel Symposia book series (ABEL, volume 15)

Abstract

We present in this paper an application of the theory of principal bundles to the problem of nonlinear dimensionality reduction in data analysis. More explicitly, we derive, from a 1-dimensional persistent cohomology computation, explicit formulas for circle-valued functions on data with nontrivial underlying topology. We show that the language of principal bundles leads to coordinates defined on an open neighborhood of the data, but computed using only a smaller subset of landmarks. It is in this sense that the coordinates are sparse. Several data examples are presented, as well as theoretical results underlying the construction.

Notes

Acknowledgements

This work was partially supported by the NSF under grant DMS-1622301 and DARPA under grant HR0011-16-2-003.

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Computational Mathematics, Science and EngineeringMichigan State UniversityEast LansingUSA
  2. 2.Department of MathematicsMichigan State UniversityEast LansingUSA

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