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Persistent Cohomology and Circular Coordinates

Discrete & Computational Geometry volume 45, pages 737–759 (2011)Cite this article

Abstract

Nonlinear dimensionality reduction (NLDR) algorithms such as Isomap, LLE, and Laplacian Eigenmaps address the problem of representing high-dimensional nonlinear data in terms of low-dimensional coordinates which represent the intrinsic structure of the data. This paradigm incorporates the assumption that real-valued coordinates provide a rich enough class of functions to represent the data faithfully and efficiently. On the other hand, there are simple structures which challenge this assumption: the circle, for example, is one-dimensional, but its faithful representation requires two real coordinates. In this work, we present a strategy for constructing circle-valued functions on a statistical data set. We develop a machinery of persistent cohomology to identify candidates for significant circle-structures in the data, and we use harmonic smoothing and integration to obtain the circle-valued coordinate functions themselves. We suggest that this enriched class of coordinate functions permits a precise NLDR analysis of a broader range of realistic data sets.

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Authors and Affiliations

  1. Department of Mathematics, Pomona College, Claremont, CA, USA

    Vin de Silva

  2. Departments of Computer Science and Mathematics, Stanford University, Stanford, CA, USA

    Dmitriy Morozov

  3. Department of Mathematics, Stanford University, Stanford, CA, USA

    Mikael Vejdemo-Johansson

Authors
  1. Vin de Silva
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  2. Dmitriy Morozov
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  3. Mikael Vejdemo-Johansson
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Corresponding author

Correspondence to Mikael Vejdemo-Johansson.

Additional information

V. de Silva was partially supported by DARPA, through grants HR0011-05-1-0007 (TDA) and HR0011-07-1-0002 (SToMP). The author holds a Digiteo Chair.

M. Vejdemo-Johansson was partially supported by the Office of Naval Research, through grant N00014-08-1-0931.

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de Silva, V., Morozov, D. & Vejdemo-Johansson, M. Persistent Cohomology and Circular Coordinates. Discrete Comput Geom 45, 737–759 (2011). https://doi.org/10.1007/s00454-011-9344-x

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  • DOI: https://doi.org/10.1007/s00454-011-9344-x

Keywords

  • Dimensionality reduction
  • Computational topology
  • Persistent homology
  • Persistent cohomology