text: 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.
  @cite_8: 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.
  @cite_7: 
  @cite_11: Numerous methods or algorithms have been designed to solve the problem of nonlinear dimensionality reduction (NLDR). However, very few among them are able to embed efficiently 'circular' manifolds like cylinders or tori, which have one or more essential loops. This paper presents a simple and fast procedure that can tear or cut those manifolds, i.e. break their essential loops, in order to make their embedding in a low-dimensional space easier. The key idea is the following: starting from the available data points, the tearing procedure represents the underlying manifold by a graph and then builds a maximum subgraph with no loops anymore. Because it works with a graph, the procedure can preprocess data for all NLDR techniques that uses the same representation. Recent techniques using geodesic distances (Isomap, geodesic Sammon's mapping, geodesic CCA, etc.) or K-ary neighborhoods (LLE, hLLE, Laplacian eigenmaps) fall in that category. After describing the tearing procedure in details, the paper comments a few experimental results. (c) 2005 Elsevier B.V. All rights reserved.
summary: There have been other attempts to address the problem of finding good coordinate representations of simple non-Euclidean data spaces. One approach @cite_7 is to use modified versions of multidimensional scaling specifically devised to find the best embedding of a data set into the cylinder, the sphere and so on. The target space has to be chosen in advance. Another class of approaches @cite_11 @cite_8 involves cutting the data manifold along arcs and curves until it has trivial topology. The resulting configuration can then be embedded in Euclidean space in the usual way. In our approach, the number of circular coordinates is not fixed in advance, but is determined experimentally after a persistent homology calculation. Moreover, there is no cutting involved; the coordinate functions respect the original topology of the data.