Persistent Cohomology and Circular Coordinates |
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Authors: | Vin?de?Silva Dmitriy?Morozov Email author" target="_blank">Mikael?Vejdemo-JohanssonEmail author |
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Institution: | 1.Department of Mathematics,Pomona College,Claremont,USA;2.Departments of Computer Science and Mathematics,Stanford University,Stanford,USA;3.Department of Mathematics,Stanford University,Stanford,USA |
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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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Keywords: | |
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