Random action of compact Lie groups and minimax estimation of a mean pattern

TL;DR

This paper develops a method for estimating a mean pattern in data affected by shape variability modeled through compact Lie group actions, using Fourier deconvolution and harmonic analysis, and analyzes the minimax risk in an asymptotic setting.

Contribution

It introduces a novel approach to mean pattern estimation under Lie group actions using harmonic analysis and derives minimax risk bounds in a nonparametric deconvolution framework.

Findings

Estimation problem reduces to deconvolution over Lie groups.

Derived upper and lower bounds for minimax quadratic risk.

Risk rate depends on the smoothness of the density of group elements.

Abstract

This paper considers the problem of estimating a mean pattern in the setting of Grenander's pattern theory. Shape variability in a data set 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 Grenander's 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.