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Persistent cohomology and circular coordinates
Department of Mathematics, Stanford University, California.ORCID iD: 0000-0001-6322-7542
2009 (English)In: Proceedings of the 25th annual symposium on Computational geometry, Association for Computing Machinery (ACM), 2009, 227-236 p.Conference paper (Refereed)
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
Association for Computing Machinery (ACM), 2009. 227-236 p.
Keyword [en]
dimensionality reduction, computational topology, persistent homology, persistent cohomology
National Category
Computational Mathematics
URN: urn:nbn:se:kth:diva-150364DOI: 10.1145/1542362.1542406ISI: 000267982900035ScopusID: 2-s2.0-70849131458OAI: diva2:742583
SoCG '09 25th Annual Symposium on Computational Geometry, Aarhus, Denmark — June 08 - 10, 2009

QC 20140903

Available from: 2014-09-02 Created: 2014-09-02 Last updated: 2014-12-16Bibliographically approved

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Vejdemo-Johansson, Mikael
Computational Mathematics

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