Fast manifold learning based on riemannian normal coordinates

Authors:
Anders Brun;Carl-Fredrik Westin;Magnus Herberthson;Hans Knutsson
Affiliations:
Department of Biomedical Engineering, Linköpings Universitet, Linköping, Sweden;Laboratory of Mathematics in Imaging, Harvard Medical School, Boston, MA;Department of Mathematics, Linköpings universitet, Linköping, Sweden;Department of Biomedical Engineering, Linköpings Universitet, Linköping, Sweden
Venue:
SCIA'05 Proceedings of the 14th Scandinavian conference on Image Analysis
Year:
2005

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Abstract

We present a novel method for manifold learning, i.e. identification of the low-dimensional manifold-like structure present in a set of data points in a possibly high-dimensional space. The main idea is derived from the concept of Riemannian normal coordinates. This coordinate system is in a way a generalization of Cartesian coordinates in Euclidean space. We translate this idea to a cloud of data points in order to perform dimension reduction. Our implementation currently uses Dijkstra's algorithm for shortest paths in graphs and some basic concepts from differential geometry. We expect this approach to open up new possibilities for analysis of e.g. shape in medical imaging and signal processing of manifold-valued signals, where the coordinate system is “learned” from experimental high-dimensional data rather than defined analytically using e.g. models based on Lie-groups.