Persistent Cohomology and Circular Coordinates

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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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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Keywords

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