Persistent cohomology and circular coordinates

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 p...

• Type: Conference
• Year: 2009
• Where: COMPGEOM
• Authors: Vin de Silva, Mikael Vejdemo-Johansson
• Keywords: COMPGEOM 2009, Dimensionality Reduction, Discrete Geometry, Nonlinear Dimensionality Reduction, Persistent Cohomology