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Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations

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Abstract

We present a framework which leverages the underlying topology of a data set, in order to produce appropriate coordinate representations. In particular, we show how to construct maps to real and complex projective spaces, given appropriate persistent cohomology classes. An initial map is obtained in two steps: First, the persistent cohomology of a sparse filtration is used to compute systems of transition functions for (real and complex) line bundles over neighborhoods of the data. Next, the transition functions are used to produce explicit classifying maps for the induced bundles. A framework for dimensionality reduction in projective space (Principal Projective Components) is also developed, aimed at decreasing the target dimension of the original map. Several examples are provided as well as theorems addressing choices in the construction.

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Notes

  1. Using a 7th nearest neighbor graph.

  2. Determined experimentally using a persistent cohomology computation.

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Acknowledgements

The author would like to thank Nils Baas, Ulrich Bauer, John Harer, Dmitriy Morozov and Don Sheehy for extremely helpful conversations regarding the contents of this paper. The detailed comments of the anonymous reviewers were invaluable in fixing multiple imprecisions found in the original draft; thank you for the high-quality feedback. This work was partially supported by the NSF under Grant DMS-1622301 and DARPA under Grant HR0011-16-2-003.

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  1. Department of Computational Mathematics, Science & Engineering and Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Jose A. Perea

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Correspondence to Jose A. Perea.

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Perea, J.A. Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations. Discrete Comput Geom 59, 175–225 (2018). https://doi.org/10.1007/s00454-017-9927-2

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  • DOI: https://doi.org/10.1007/s00454-017-9927-2

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