inconsistency of template estimation by minimizing of the variance/pre-variance in the quotient space

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ID: 190842
2017
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Abstract
We tackle the problem of template estimation when data have been randomly deformed under a group action in the presence of noise. In order to estimate the template, one often minimizes the variance when the influence of the transformations have been removed (computation of the Fréchet mean in the quotient space). The consistency bias is defined as the distance (possibly zero) between the orbit of the template and the orbit of one element which minimizes the variance. In the first part, we restrict ourselves to isometric group action, in this case the Hilbertian distance is invariant under the group action. We establish an asymptotic behavior of the consistency bias which is linear with respect to the noise level. As a result the inconsistency is unavoidable as soon as the noise is enough. In practice, template estimation with a finite sample is often done with an algorithm called “max-max”. In the second part, also in the case of isometric group finite, we show the convergence of this algorithm to an empirical Karcher mean. Our numerical experiments show that the bias observed in practice can not be attributed to the small sample size or to a convergence problem but is indeed due to the previously studied inconsistency. In a third part, we also present some insights of the case of a non invariant distance with respect to the group action. We will see that the inconsistency still holds as soon as the noise level is large enough. Moreover we prove the inconsistency even when a regularization term is added.
Reference Key
devilliers2017entropyinconsistency Use this key to autocite in the manuscript while using SciMatic Manuscript Manager or Thesis Manager
Authors ;Loïc Devilliers;Stéphanie Allassonnière;Alain Trouvé;Xavier Pennec
Journal European journal of medicinal chemistry
Year 2017
DOI
10.3390/e19060288
URL
http://www.mdpi.com/1099-4300/19/6/288
Keywords Keywords not found

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