动物运动子流形估计的学习理论
Learning Theory for Estimation of Animal Motion Submanifolds
摘要：
本文介绍了一种新的方法，估计和近似的动物运动的子流形模型的制定和实验测试。假设动物运动支持在一个构形流形$Q$上，该构形流形是欧氏空间$X\approx \mathbb{R}^d$的一个光滑、连通、规则嵌入的黎曼子流形，并且该流形$Q$同胚于一个已知的光滑黎曼流形$S$。流形的估计是通过找到一个未知的映射$γ:S\rightarrow Q\subset X$，将流形$S$映射到$Q$。整个问题是作为一个分布免费学习问题的测量$\mathbb{Z}=S\times X$的流形。也就是说，假设实验生成根据$\mathbb{Z}$上的未知概率密度$μ$生成的样本的有限集合$\{(s_i,x_i)\}_{i=1}^m\subset \mathbb{Z}^m$。本文推导出$γ$的近似$γ_{n,m}$，它是基于$m$样本的，并且包含在近似的$N(n)$维空间中。本文定义的充分条件表明，在$L^2_μ(S)$的收敛速度对应于那些已知的经典分布自由学习理论在欧几里德空间。具体而言，本文推导出的充分条件，保证收敛速度的形式$$\mathbb{E} \left (\|γ_μ^j-γ_{n,m}^j\|_{L^2_μ(S)}^2\right )\leq C_1 N(n)^{-r} + C_2 \frac{N(n)\log(N(n))}{m}$$为常数$C_1,C_2$与$γ_μ:=\{γ^1_μ,\ldots,γ^d_μ\}$的回归函数$γ_μ:S\rightarrow Q\subset X$和$γ_{n,m}:=\{γ^1_{n,j},\ldots,γ^d_{n,m}\}$。
英文摘要：
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 $γ_{n,m}:=\{γ^1_{n,j},\ldots,γ^d_{n,m}\}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
