Learning Theory for Estimation of Animal Motion Submanifolds
This work addresses the challenge of modeling animal motion for researchers in computational biology or robotics, but it appears incremental as it extends existing learning theory to a manifold setting without claiming broad SOTA impact.
The paper tackles the problem of estimating submanifold models for animal motion by formulating it as a distribution-free learning problem, deriving approximations with convergence rates similar to classical learning theory in Euclidean space, specifically bounding the expected squared error with terms involving sample size and approximation dimension.
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 $γ:S\rightarrow Q\subset X$ 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 $μ$ on $\mathbb{Z}$. This paper derives approximations $γ_{n,m}$ of $γ$ 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 $L^2_μ(S)$ 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 $$\mathbb{E} \left (\|γ_μ^j-γ_{n,m}^j\|_{L^2_μ(S)}^2\right )\leq C_1 N(n)^{-r} + C_2 \frac{N(n)\log(N(n))}{m}$$for constants $C_1,C_2$ with $γ_μ:=\{γ^1_μ,\ldots,γ^d_μ\}$ the regressor function $γ_μ:S\rightarrow Q\subset X$ and $γ_{n,m}:=\{γ^1_{n,j},\ldots,γ^d_{n,m}\}$.