Andrew J. Kurdila

Andrew J. Kurdila, Parag Bobade

Koopman theory studies dynamical systems in terms of operator theoretic properties of the Perron-Frobenius operator $\mathcal{P}$ and Koopman operator $\mathcal{U}$ respectively. In this paper, we derive the rates of convergence of approximations of $\mathcal{P}$ or $\mathcal{U}$ that are generated by finite dimensional bases like wavelets, multiwavelets, and eigenfunctions, as well as approaches that use samples of the input and output of the system in conjunction with these bases. We introduce a general class of priors that describe the information available for constructing such approximations and facilitate the error estimates in many applications of interest. These priors are defined in terms of the action of $\mathcal{P}$ or $\mathcal{U}$ on certain linear approximation spaces. The rates of convergence for the estimates of these operators are investigated under a variety of situations that are motivated from associated assumptions in practical applications. When the estimates of these operators are generated by samples, it is shown that the error in approximation of the Perron-Frobenius or Koopman operators can be decomposed into two parts, the approximation error and the sampling error. This result emphasizes that sample-based estimates of Perron-Frobenius and Koopman operators are subject to the well-known trade-off between the bias and variance that contribute to the error, a balance that also features in nonlinear regression and statistical learning theory.

Ali Bouland, Shengyuan Niu, Sai Tej Paruchuri et al.

This paper studies convergence rates for some value function approximations that arise in a collection of reproducing kernel Hilbert spaces (RKHS) $H(Ω)$. By casting an optimal control problem in a specific class of native spaces, strong rates of convergence are derived for the operator equation that enables offline approximations that appear in policy iteration. Explicit upper bounds on error in value function and controller approximations are derived in terms of power function $\mathcal{P}_{H,N}$ for the space of finite dimensional approximants $H_N$ in the native space $H(Ω)$. These bounds are geometric in nature and refine some well-known, now classical results concerning convergence of approximations of value functions.

Nathan Powell, Bowei Liu, Jia Guo et al.

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.

Nathan Powell, Andrew Kurdila

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 $γ_{n,m}:=\{γ^1_{n,j},\ldots,γ^d_{n,m}\}$.

Parag Bobade, Suprotim Majumdar, Savio Pereira et al.

This paper extends a conventional, general framework for online adaptive estimation problems for systems governed by unknown nonlinear ordinary differential equations. The central feature of the theory introduced in this paper represents the unknown function as a member of a reproducing kernel Hilbert space (RKHS) and defines a distributed parameter system (DPS) that governs state estimates and estimates of the unknown function. This paper 1) derives sufficient conditions for the existence and stability of the infinite dimensional online estimation problem, 2) derives existence and stability of finite dimensional approximations of the infinite dimensional approximations, and 3) determines sufficient conditions for the convergence of finite dimensional approximations to the infinite dimensional online estimates. A new condition for persistency of excitation in a RKHS in terms of its evaluation functionals is introduced in the paper that enables proof of convergence of the finite dimensional approximations of the unknown function in the RKHS. This paper studies two particular choices of the RKHS, those that are generated by exponential functions and those that are generated by multiscale kernels defined from a multiresolution analysis.