Koopman Methods for Estimation of Motion over Unknown, Regularly Embedded Submanifolds

This paper introduces a data-dependent approximation of the forward kinematics map for certain types of motion models, those that are generated for example in applications to humans and other animals. 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.