Learning Theory for Estimation of Animal Motion Submanifolds

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 \(X\approx \mathbb{R}^d\) 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 \(\gamma:S\rightarrow Q\subset X\) that maps the manifold \(S\) into \(Q\). The overall problem is cast as a distribution-free learning problem over the manifold of measurements \(\mathbb{Z}=S\times X\). That is, it is assumed that experiments generate a finite sets \(\{(s_i,x_i)\}_{i=1}^m\subset \mathbb{Z}^m\) of samples that are generated according to an unknown probability density \(\mu\) on \(\mathbb{Z}\). This paper derives approximations \(\gamma_{n,m}\) of \(\gamma\) 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 \(L^2_\mu(S)\) 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 $$\mathbb{E} \left (\|\gamma_\mu^j-\gamma_{n,m}^j\|_{L^2_\mu(S)}^2\right )\leq C_1 N(n)^{-r} + C_2 \frac{N(n)\log(N(n))}{m}$$for constants \(C_1,C_2\) with \(\gamma_\mu:=\{\gamma^1_\mu,\ldots,\gamma^d_\mu\}\) the regressor function \(\gamma_\mu:S\rightarrow Q\subset X\) and \(\gamma_{n,m}:=\{\gamma^1_{n,j},\ldots,\gamma^d_{n,m}\}\).