Learning Theory for Estimation of Animal Motion Submanifolds

Abstract

This paper describes the formulation and experimental testing of a novel method for the estimation and approximation of submanifold models of animal motion. It is assumed that the animal motion is supported on a configuration manifold Q that is a smooth, connected, regularly embedded Riemannian submanifold of Euclidean space X R^d for some d>0, and that the manifold Q is homeomorphic to a known smooth, Riemannian manifold S. Estimation of the manifold is achieved by finding an unknown mapping :S Q X that maps the manifold S into Q. The overall problem is cast as a distribution-free learning problem over the manifold of measurements Z=S X. That is, it is assumed that experiments generate a finite sets \(s_i,x_i)\_i=1^m Z^m of samples that are generated according to an unknown probability density on Z. This paper derives approximations _n,m of that are based on the m samples and are contained in an N(n) dimensional space of approximants. The paper defines sufficient conditions that shows that the rates of convergence in L^2_(S) correspond to those known for classical distribution-free learning theory over Euclidean space. Specifically, the paper derives sufficient conditions that guarantee rates of convergence that have the form for constants C_1,C_2 with _:=\^1_,,^d_\ the regressor function _:S Q X and _n,m:=\^1_n,j,,^d_n,m\.

Tasks

Reproductions