Fast Manifold Learning Based on Riemannian Normal Coordinates

  • Anders Brun
  • Carl-Fredrik Westin
  • Magnus Herberthson
  • Hans Knutsson
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3540)

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.

Keywords

Image Patch Geodesic Distance Klein Bottle Nonlinear Dimensionality Reduction Swiss Roll 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Anders Brun
    • 1
    • 3
  • Carl-Fredrik Westin
    • 3
  • Magnus Herberthson
    • 2
  • Hans Knutsson
    • 1
  1. 1.Department of Biomedical EngineeringLinköpings UniversitetLinköpingSweden
  2. 2.Department of MathematicsLinköpings universitetLinköpingSweden
  3. 3.Laboratory of Mathematics in ImagingHarvard Medical SchoolBostonUSA

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