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Persistent cohomology and circular coordinates
Stanford University, USA.ORCID iD: 0000-0001-6322-7542
2011 (English)In: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444, Vol. 45, 737-759 p.Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
Springer Publishing Company, 2011. Vol. 45, 737-759 p.
Keyword [en]
Dimensionality reduction; Computational topology; Persistent homology; Persistent cohomology
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-150355DOI: 10.1007/s00454-011-9344-xISI: 000289521700007Scopus ID: 2-s2.0-79954629188OAI: oai:DiVA.org:kth-150355DiVA: diva2:742600
Note

QC 20140908

Available from: 2014-09-02 Created: 2014-09-02 Last updated: 2017-12-05Bibliographically approved

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