The Annals of Applied Statistics

Principal arc analysis on direct product manifolds

Sungkyu Jung, Mark Foskey, and J. S. Marron

Full-text: Open access

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.

Article information

Source
Ann. Appl. Stat., Volume 5, Number 1 (2011), 578-603.

Dates
First available in Project Euclid: 21 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.aoas/1300715203

Digital Object Identifier
doi:10.1214/10-AOAS370

Mathematical Reviews number (MathSciNet)
MR2810410

Zentralblatt MATH identifier
1220.62077

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

Citation

Jung, Sungkyu; Foskey, Mark; Marron, J. S. Principal arc analysis on direct product manifolds. Ann. Appl. Stat. 5 (2011), no. 1, 578--603. doi:10.1214/10-AOAS370. https://projecteuclid.org/euclid.aoas/1300715203


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