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 $\gamma_{ n,m}:=\{ \gamma^1_{ n,j},\ldots,\gamma^d_{ n,m}\}$.
Categories: stat.ML math.DS
