商空间中模板估计与Fr{é}chet均值的不一致性
Inconsistency of Template Estimation with the Fr{é}chet mean in Quotient Space
摘要：
我们解决了模板估计的问题时，数据已被随机变换下的等距群作用在噪声的存在下。为了估计模板，当变换的影响已经被移除时，人们通常最小化方差（在商空间中计算Fr{é}chet均值）。一致性偏差被定义为模板的轨道和使方差最小化的一个元素的轨道之间的距离（可能为零）。在这篇文章中，我们建立了一个渐近行为的一致性偏差相对于噪声水平。该行为相对于噪声水平是线性的。因此，一旦噪声足够大，不一致性就不可避免。在实践中，有限样本的模板估计通常使用称为max-max的算法来完成。我们证明了该算法的收敛性的经验Karcher平均。最后，我们的数值实验表明，在实践中观察到的偏差不能归因于小样本量或收敛问题，但确实是由于以前研究的不一致。
英文摘要：
We tackle the problem of template estimation when data have been randomly transformed under an isometric 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 Fr{é}chet mean in 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 this article we establish an asymptotic behavior of the consistency bias with respect to the noise level. This behavior is linear with respect to the noise level. As a result the inconsistency is unavoidable as soon as the noise is large enough. In practice, the template estimation with a finite sample is often done with an algorithm called max-max. We show the convergence of this algorithm to an empirical Karcher mean. Finally, our numerical experiments show that the bias observed in practice cannot be attributed to the small sample size or to a convergence problem but is indeed due to the previously studied inconsistency.
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