4 research outputs found
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 permits a precise NLDR analysis of a broader range of realistic data
sets.Comment: 10 pages, 7 figures. To appear in the proceedings of the ACM
Symposium on Computational Geometry 200
Finding Minimal Parameterizations of Cylindrical Image Manifolds
Manifold learning has become an important tool to characterize high-dimensional data that vary nonlinearly due to a few parameters. Applications to the analysis of medical imagery and human motion patterns have been successful despite the lack of effective tools to parameterize cyclic data sets. This paper offers an initial approach to this problem, and provides for a minimal parameterization of points that are drawn from cylindrical manifolds—data whose (unknown) generative model includes a cyclic and a non-cyclic parameter. Solving for this special case is important for a number of current, practical applications and provides a start toward a general approach to cyclic manifolds. We offer results on synthetic and real data sets and illustrate an application to de-noising cardiac ultrasound images. 1