未知子流形上动物运动估计的Koopman方法
Koopman Methods for Estimation of Animal Motions over Unknown Submanifolds
摘要：
本文介绍了一种依赖于数据的近似的正向运动学映射的某些类型的动物运动模型。假设运动支持在低维、未知配置流形$Q$上，该流形规则地嵌入在高维欧氏空间$X:=\mathbb{R}^d$中。本文介绍了一种从未知配置子流形$Q$到$n$维欧几里得观测空间$Y:=\mathbb{R}^n$的正向运动学估计方法。根据已知的核函数在周围空间$X$上定义已知的再现核希尔伯特空间（RKHS），并且使用在周围空间$X$上定义的已知核来执行计算。估计值是使用Koopman算子的某种依赖于数据的近似值来构造的，该近似值是根据$X$上的已知内核来定义的。然而，收敛速度的近似研究的限制空间中的未知流形$Q$。强的收敛速度推导出的填充距离的样本在未知的配置流形，提供了一个新的规律性结果，Koopman运营商。此外，我们表明，推导出的收敛速度可以在某些情况下适用于估计所产生的扩展动态模式分解（EDMD）方法。我们说明了模拟数据的估计特性，以及在运动捕捉实验中收集的样本。
英文摘要：
This paper introduces a data-dependent approximation of the forward kinematics map for certain types of animal motion models. It is assumed that motions are supported on a low-dimensional, unknown configuration manifold $Q$ that is regularly embedded in high dimensional Euclidean space $X:=\mathbb{R}^d$. This paper introduces a method to estimate forward kinematics from the unknown configuration submanifold $Q$ to an $n$-dimensional Euclidean space $Y:=\mathbb{R}^n$ of observations. A known reproducing kernel Hilbert space (RKHS) is defined over the ambient space $X$ in terms of a known kernel function, and computations are performed using the known kernel defined on the ambient space $X$. Estimates are constructed using a certain data-dependent approximation of the Koopman operator defined in terms of the known kernel on $X$. However, the rate of convergence of approximations is studied in the space of restrictions to the unknown manifold $Q$. Strong rates of convergence are derived in terms of the fill distance of samples in the unknown configuration manifold, provided that a novel regularity result holds for the Koopman operator. Additionally, we show that the derived rates of convergence can be applied in some cases to estimates generated by the extended dynamic mode decomposition (EDMD) method. We illustrate characteristics of the estimates for simulated data as well as samples collected during motion capture experiments.
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。
