• DOI:10.1109/TIT.2012.2185484
• Corpus ID: 4803960

Random Action of Compact Lie Groups and Minimax Estimation of a Mean Pattern

@article{Bigot2012RandomAO,
  title={Random Action of Compact Lie Groups and Minimax Estimation of a Mean Pattern},
  author={J{\'e}r{\'e}mie Bigot and Claire Christophe and S{\'e}bastien Gadat},
  journal={IEEE Transactions on Information Theory},
  year={2012},
  volume={58},
  pages={3509-3520}
}

• Jérémie Bigot, Claire Christophe, S. Gadat
• Published 13 October 2011
• Mathematics
• IEEE Transactions on Information Theory

This paper considers the problem of estimating a mean pattern in the setting of Grenander's pattern theory. Shape variability in a dataset of curves or images is modeled by the random action of elements in a compact Lie group on an infinite dimensional space. In the case of observations contaminated by an additive Gaussian white noise, it is shown that estimating a reference template in the setting of Grenanders pattern theory falls into the category of deconvolution problems over Lie groups. To obtain this result, we build an estimator of a mean pattern by using Fourier deconvolution and harmonic analysis on compact Lie groups. In an asymptotic setting where the number of observed curves or images tends to infinity, we derive upper and lower bounds for the minimax quadratic risk over Sobolev balls. This rate depends on the smoothness of the density of the random Lie group elements representing shape variability in the data, which makes a connection between estimating a mean pattern and standard deconvolution problems in nonparametric statistics.