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This paper describes the formulation and experimental testing of a novelmethod for the estimation and approximation of submanifold models of animalmotion. It is assumed that the animal motion is supported on a configurationmanifold $Q$ that is a smooth, connected, regularly embedded Riemanniansubmanifold of Euclidean space $X\approx \mathbb{R}^d$ for some $d>0$, and thatthe 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 $Q$. Theoverall problem is cast as a distribution-free learning problem over themanifold of measurements $\mathbb{Z}=S\times X$. That is, it is assumed thatexperiments 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 $N(n)$ dimensionalspace of approximants. The paper defines sufficient conditions that shows thatthe rates of convergence in $L^2_\mu(S)$ correspond to those known forclassical distribution-free learning theory over Euclidean space. Specifically,the paper derives sufficient conditions that guarantee rates of convergencethat 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$\gamma_{n,m}:=\{\gamma^1_{n,j},\ldots,\gamma^d_{n,m}\}$.

Learning Theory for Estimation of Animal Motion Submanifolds