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March 2011 Principal arc analysis on direct product manifolds
Sungkyu Jung, Mark Foskey, J. S. Marron
Ann. Appl. Stat. 5(1): 578-603 (March 2011). DOI: 10.1214/10-AOAS370

Abstract

We propose a new approach to analyze data that naturally lie on manifolds. We focus on a special class of manifolds, called direct product manifolds, whose intrinsic dimension could be very high. Our method finds a low-dimensional representation of the manifold that can be used to find and visualize the principal modes of variation of the data, as Principal Component Analysis (PCA) does in linear spaces. The proposed method improves upon earlier manifold extensions of PCA by more concisely capturing important nonlinear modes. For the special case of data on a sphere, variation following nongeodesic arcs is captured in a single mode, compared to the two modes needed by previous methods. Several computational and statistical challenges are resolved. The development on spheres forms the basis of principal arc analysis on more complicated manifolds. The benefits of the method are illustrated by a data example using medial representations in image analysis.

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Sungkyu Jung. Mark Foskey. J. S. Marron. "Principal arc analysis on direct product manifolds." Ann. Appl. Stat. 5 (1) 578 - 603, March 2011. https://doi.org/10.1214/10-AOAS370

Information

Published: March 2011
First available in Project Euclid: 21 March 2011

zbMATH: 1220.62077
MathSciNet: MR2810410
Digital Object Identifier: 10.1214/10-AOAS370

Keywords: directional data , folded Normal distribution , image analysis , Manifold , medial representation , nonlinear dimension reduction , Principal Component Analysis

Rights: Copyright © 2011 Institute of Mathematical Statistics

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Vol.5 • No. 1 • March 2011
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