期刊
DISCRETE & COMPUTATIONAL GEOMETRY
卷 45, 期 4, 页码 737-759出版社
SPRINGER
DOI: 10.1007/s00454-011-9344-x
关键词
Dimensionality reduction; Computational topology; Persistent homology; Persistent cohomology
资金
- DARPA [HR0011-05-1-0007, HR0011-07-1-0002]
- Office of Naval Research [N00014-08-1-0931]
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.
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