Persistent Cohomology and Circular Coordinates

TL;DR

This paper introduces a method using persistent cohomology to identify and construct circle-valued coordinates for nonlinear dimensionality reduction, enabling better representation of data with circular structures.

Contribution

It develops a novel approach combining persistent cohomology and harmonic smoothing to produce circle-valued coordinates for complex data sets.

Findings

Successfully identifies significant circular structures in data

Enables more accurate low-dimensional representations of nonlinear data

Broadens the applicability of NLDR methods to circular data

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.