On Omitting and Hitting Properties for Means on Circles and Shape Spaces
Stephan Huckemann (May 24, 2012)
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The classical central limit theorem states that suitably translated and root n rescaled independent sample means tend to a multivariate Gaussian. Under certain, still rather restrictive conditions, it has been shown by Bhattacharya and Patrangenaru (2005) that the analog holds true on manifolds. One condition, namely uniqueness has been pushed to "data contained in a geodesic half ball" by Afsari (2011), which in particular encompasses "omitting a neighborhood of the cut locus" if non-void.
Determining asymptotics when the cut locus is not omitted proves to be challenging. For circles we present an exhaustive treatment of uniqueness and, in view of asymptotics, of the role of mass around the antipodal point.
Another issue turning up in shape spaces -- which may be manifolds with singularities -- is whether means omit these singularities and are stably assumed on the manifold part. We show that while intrinsic and Ziezold means are manifold stable, Procrustes means may hit singularities.
In consequence, e.g. for 3D shape analysis, given uniqueness, discrimination and classification based on the two-sample test is possible for intrinsic or Ziezold means. Procrustes means, however, may disqualify.
This talk is based on joint work with Thomas Hotz.