Persistent Cohomology and Circular Coordinates

• Published in: Discrete & computational geometry Vol. 45; no. 4; pp. 737 - 759
• Main Authors: de Silva, Vin, Morozov, Dmitriy, Vejdemo-Johansson, Mikael
• Format: Journal Article
• Language: English
• Published: New York Springer-Verlag 01.06.2011; Springer Nature B.V
• Subjects: Algorithms; Combinatorics; Computational Mathematics and Numerical Analysis; Coordinate transformations; Geometry; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Nonlinearity; Representations; Smoothing; Strategy; Dimensionality reduction; Persistent homology; Persistent cohomology; Computational topology
• ISSN: 0179-5376; 1432-0444; 1432-0444
• DOI: 10.1007/s00454-011-9344-x

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.