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On Means and Their Asymptotics: Circles and Shape Spaces

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Abstract

We survey some effects that singular strata may have in the positive curvature context of circles and shape spaces when conducting (semi-)intrinsic statistical analyses. Here, the analysis of data on a stratified space is based on statistical descriptors defined in a possibly different stratified space. E.g. in geodesic principal component analysis for shape spaces, shape data are described by generalized geodesics which naturally form a shape space of their own, different from the original one. In a general context, if the descriptors are obtained as generalized Fréchet means, under rather general circumstances, a strong law of large numbers is valid. If furthermore the descriptors are sufficiently well behaved, a classical central limit theorem can be adopted. One of the crucial conditions is that hitting of singular strata as well as of cut loci, if present, must be controlled. We review the statistical role of the cut locus of intrinsic means for circles as well as that of singular strata for shape spaces (occurring where the group action is degenerate) and conclude with an identification of potential research directions.

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Author information

Authors and Affiliations

  1. Institute for Mathematical Stochastics, University of Göttingen, Goldschmidtstrasse 7, 37077, Göttingen, Germany

    Stephan Huckemann

  2. Mathematics Institute, TU Ilmenau, Weimarer Str. 25, 98693, Ilmenau, Germany

    Thomas Hotz

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  1. Stephan Huckemann
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  2. Thomas Hotz
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Correspondence to Stephan Huckemann.

Additional information

S. Huckemann’s work was supported by DFG HU 1575/2-1 and the Niedersachsen Vorab of the Volkswagen Foundation.

T. Hotz’s work was supported by DFG CRC 803.

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Huckemann, S., Hotz, T. On Means and Their Asymptotics: Circles and Shape Spaces. J Math Imaging Vis 50, 98–106 (2014). https://doi.org/10.1007/s10851-013-0462-3

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