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Fractional Order System Identification With Occupation kernel Regression

IEEE CONTROL SYSTEMS LETTERS 2022年 6卷 19-24页

作者： Li, Xiuying Rosenfeld, Joel A. Changshu Inst Technol Dept Math Suzhou 215500 Peoples R China Univ S Florida Dept Math & Stat Tampa FL 33620 USA

While fractional order systems have been employed broadly throughout science and engineering, system identification aimed at nonlinear fractional order dynamical systems remain in their infancy. One reason for this is that local estimates cannot be used to obtain a sample of fractional order dynamics in the same way that is done for integer order systems. This letter leverages occupation kernels to poise a trajectory as the fundamental unit of data from a fractional order dynamical system. When combined with a regularized regression problem, an approximation of fractional order dynamics is obtained as a linear combination of occupation kernels. A battery of numerical experiments are executed to validate the developed method, and it is demonstrated over two dynamical systems that accurate estimates of fractional order dynamics can be obtained both along trajectories and also nearby regions.

关键词： kernel Trajectory hilbert space Nonlinear dynamical systems Estimation Tools Linear systems Fractional order differential equations numerical methods occupation kernels reproducing kernel hilbert spaces system identification

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A converse sampling theorem in reproducing kernel Banach spaces

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SAMPLING THEORY SIGNAL PROCESSING AND DATA ANALYSIS 2022年第2期20卷 1-19页

作者： Centeno, Hernan Medina, Juan Miguel Univ Buenos Aires Fac Ingn Dept Matemat Paseo Colon 850 RA-1063 Buenos Aires Argentina Univ Buenos Aires Fac Ingn Dept Matemat Paseo Colon 850 RA-1063 Buenos Aires Argentina Consejo Nacl Invest Cient & Tecn IAM Paseo Colon 850 RA-1063 Buenos Aires Argentina Pontificia Univ Catolica Argentina LaBIS Buenos Aires Argentina

We present a converse Kramer type sampling theorem over semi-inner product reproducing kernel Banach spaces. Assuming that a sampling expansion holds for every f belonging to a semi-inner product reproducing kernel Banach space B for a fixed sequence of interpolating functions {a(j)(-1)S(j)(t)}(j )and a subset of sampling points {t(j)}(j), it results that such sequence must be a X-d(& lowast;)-Riesz basis and a sampling basis for the space. Moreover, there exists an equivalent (in norm) reproducing kernel Banach space with a reproducing kernel G(samp) such that {a(-1)(j)G(samp)(t(j),.)}(j) and {a(j)(-1)S(j)(.)}(j) are biorthogonal. These results are a generalization of some known results over reproducing kernel hilbert spaces.

关键词： Sampling basis Non-uniform sampling reproducing kernel hilbert spaces reproducing kernel Banach spaces Kramer sampling theorems Semi-inner products Kramer sampling theorems Semi-inner products X-d-frames X-d-Riesz basis

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Training Lipschitz continuous operators using reproducing kernels 4

Training Lipschitz continuous operators using reproducing ke...

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4th Annual Conference on Learning for Dynamics and Control (L4DC)

作者： van Waarde, Henk J. Sepulchre, Rodolphe Univ Groningen Bernoulli Inst Groningen Netherlands Univ Cambridge Control Grp Cambridge England

This paper proposes that Lipschitz continuity is a natural outcome of regularized least squares in kernel-based learning. Lipschitz continuity is an important proxy for robustness of input-output operators. It is also instrumental for guaranteeing closed-loop stability of kernel-based controlllers through small incremental gain arguments. We introduce a new class of nonexpansive kernels that are shown to induce hilbert spaces consisting of only Lipschitz continuous operators. The Lipschitz constant of estimated operators within such hilbert spaces can be tuned by suitable selection of a regularization parameter. As is typical for kernel-based models, input-output operators are estimated from data by solving tractable systems of linear equations. The approach thus constitutes a promising alternative to Lipschitz-bounded neural networks, that have recently been investigated but are computationally expensive to train.

关键词： reproducing kernel hilbert spaces robustness Lipschitz continuity monotonicity

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Online regularized learning algorithm for functional data

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JOURNAL OF COMPLEXITY 2024年 82卷

作者： Mao, Yuan Guo, Zheng-Chu Zhejiang Univ Sch Math Sci Hangzhou 310058 Peoples R China

In recent years, functional linear models have attracted growing attention in statistics and machine learning for recovering the slope function or its functional predictor. This paper considers online regularized learning algorithm for functional linear models in a reproducing kernel hilbert space. It provides convergence analysis of excess prediction error and estimation error with polynomially decaying step-size and constant step-size, respectively. Fast convergence rates can be derived via a capacity dependent analysis. Introducing an explicit regularization term extends the saturation boundary of unregularized online learning algorithms with polynomially decaying step-size and achieves fast convergence rates of estimation error without capacity assumption. In contrast, the latter remains an open problem for the unregularized online learning algorithm with decaying step-size. This paper also demonstrates competitive convergence rates of both prediction error and estimation error with constant step-size compared to existing literature. (c) 2024 Elsevier Inc. All rights reserved.

关键词： Functional linear model Online learning reproducing kernel hilbert spaces Regularization

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A kernel framework for learning differential equations and their solution operators

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PHYSICA D-NONLINEAR PHENOMENA 2024年 460卷

作者： Long, Da Mrvaljevic, Nicole Zhe, Shandian Hosseini, Bamdad Univ Utah Sch Comp Salt Lake City UT 84112 USA Univ Washington Dept Appl Math Seattle WA 98195 USA

This article presents a three-step kernel framework for regression of the functional form of differential equations (DEs) and learning their solution operators. Given a training set consisting of pairs of noisy DE solutions and source/boundary terms on a mesh: (i) kernel smoothing is utilized to denoise the data and approximate derivatives of the solution;(ii) This information is then used in a kernel regression model to learn the functional form of the DE;(iii) The learned DE is then used within a numerical solver to approximate the solution of the DE with a new source term or initial data, thereby constituting an operator learning framework. Numerical experiments compare the method to state-of-the-art algorithms. In DE learning our framework matches the performance of Sparse Identification of nonlinear Dynamical Systems (SINDy) while in operator learning the method has superior performance compared to well-established neural network methods in low training data regimes.

关键词： Equation discovery Operator learning reproducing kernel hilbert spaces Physics informed machine learning

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Transporting positive definiteness

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ACTA SCIENTIARUM MATHEMATICARUM 2022年第1-2期88卷 505-514页

作者： Szafraniec, Franciszek Hugon Uniwersytet Jagiellonski Inst Matemati Ul Lojasiewicza 6 PL-30348 Krakow Poland

This is a rough guide to the topic which might be worthy to develop further on. Expanding positive definiteness beyond its presupposed scope is intriguing due to a number of possible applications. The paper though loo... 详细信息

关键词： positive definite kernels reproducing kernel hilbert spaces B. Sz.-Nagy dilation theorem Pedrick's tilde correspondence composition in reproducing kernel hilbert spaces Hamburger moment problem

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reproducing kernel hilbert space compactification of unitary evolution groups

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APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS 2021年 54卷 75-136页

作者： Das, Suddhasattwa Giannakis, Dimitrios Slawinska, Joanna NYU Courant Inst Math Sci 251 Mercer St New York NY 10012 USA Univ Helsinki Finnish Ctr Artificial Intelligence Dept Comp Sci FI-00014 Helsinki Finland

A framework for coherent pattern extraction and prediction of observables of measure-preserving, ergodic dynamical systems with both atomic and continuous spectral components is developed. This framework is based on an approximation of the generator of the system by a compact operator W-tau on a reproducing kernel hilbert space (RKHS). The operator W-tau is skew-adjoint, and thus can be represented by a projection-valued measure, discrete by compactness, with an associated orthonormal basis of eigenfunctions. These eigenfunctions are ordered in terms of a Dirichlet energy, and provide a notion of coherent observables under the dynamics akin to the Koopman eigenfunctions associated with the atomic part of the spectrum. In addition, W-tau generates a unitary evolution group {e(tW tau)}t epsilon R on the RKHS, which approximates the unitary Koopman group of the system. We establish convergence results for the spectrum and Borel functional calculus of W-tau as tau -> 0(+), as well as an associated data-driven formulation utilizing time series data. Numerical applications to ergodic systems with atomic and continuous spectra, namely a torus rotation, the Lorenz 63 system, and the Rossler system, are presented. (C) 2021 The Author(s). Published by Elsevier Inc.

关键词： Koopman operators Perron-Frobenius operators Ergodic dynamical systems reproducing kernel hilbert spaces Spectral theory

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Data Analysis from Empirical Moments and the Christoffel Function

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FOUNDATIONS OF COMPUTATIONAL MATHEMATICS 2021年第1期21卷 243-273页

作者： Pauwels, Edouard Putinar, Mihai Lasserre, Jean-Bernard Univ Toulouse CNRS AOC IRIT Toulouse France IRT St Exupery DEEL Toulouse France Univ Calif Santa Barbara Santa Barbara CA 93106 USA Newcastle Univ Newcastle Upon Tyne Tyne & Wear England Univ Toulouse CNRS LAAS 7 Ave Colonel Roche F-31077 Toulouse France Univ Toulouse LAAS Inst Math 7 Ave Colonel Roche F-31077 Toulouse France

Spectral features of the empirical moment matrix constitute a resourceful tool for unveiling properties of a cloud of points, among which, density, support and latent structures. This matrix is readily computed from an input dataset, and its eigen decomposition can then be used to identify algebraic properties of the support or density/support estimates with the Christoffel function. It is already well known that the empirical moment matrix encodes a great deal of subtle attributes of the underlying measure. Starting from this object as base of observations, we combine ideas from statistics, real algebraic geometry, orthogonal polynomials and approximation theory for opening new insights relevant for machine learning problems with data supported on algebraic sets. Refined concepts and results from real algebraic geometry and approximation theory are empowering a simple tool (the empirical moment matrix) for the task of solving non-trivial questions in data analysis. We provide (1) theoretical support, (2) numerical experiments and (3) connections to real-world data as a validation of the stamina of the empirical moment matrix approach.

关键词： Christoffel-Darboux kernel Empirical measure Support inference Manifold Density estimation reproducing kernel hilbert spaces

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Recursive estimation of conditional kernel mean embeddings

The Journal of Machine Learning Research

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The Journal of Machine Learning Research 2024年第1期25卷 12780-12814页

作者： Ambrus Tamás Balázs Csanád Csáji Institute for Computer Science and Control (SZTAKI) Hungarian Research Network (HUN-REN) Budapest Hungary Department of Probability Theory and Statistics Institute of Mathematics Eötvös Loránd University (ELTE) Budapest Hungary

kernel mean embeddings, a widely used technique in machine learning, map probability distributions to elements of a reproducing kernel hilbert space (RKHS). For supervised learning problems, where input-output pairs are observed, the conditional distribution of outputs given the inputs is a key object. The input dependent conditional distribution of an output can be encoded with an RKHS valued function, the conditional kernel mean map. In this paper we present a new recursive algorithm to estimate the conditional kernel mean map in a hilbert space valued L2 space, that is in a Bochner space. We prove the weak and strong L2 consistency of our recursive estimator under mild conditions. The idea is to generalize Stone's theorem for hilbert space valued regression in a locally compact Polish space. We present new insights about conditional kernel mean embeddings and give strong asymptotic bounds regarding the convergence of the proposed recursive method. Finally, the results are demonstrated on three application domains: for inputs coming from Euclidean spaces, Riemannian manifolds and locally compact subsets of function spaces.

关键词： conditional kernel mean embeddings recursive estimation reproducing kernel hilbert spaces strong consistency nonparametric inference Riemannian manifolds

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kernel embedding of measures and low-rank approximation of integral operators

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POSITIVITY 2024年第2期28卷 29-29页

作者： Gauthier, Bertrand Cardiff Univ Sch Math AbacwsSenghennydd Rd Cardiff CF24 4AG Wales

We describe a natural coisometry from the hilbert space of all hilbert-Schmidt operators on a separable reproducing kernel hilbert space (RKHS) H \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox { (RKHS)}\, \mathcal {H}$$\end{document} and onto the RKHS G \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}$$\end{document} associated with the squared-modulus of the reproducing kernel of H \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}$$\end{document} . Through this coisometry, trace-class integral operators defined by general measures and the reproducing kernel of H \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}$$\end{document} are isometrically represented as potentials in G \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}$$\end{document} , and the quadrature approximation of these operators is equivalent to the approximation of integral functionals on G \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{

关键词： reproducing kernel hilbert spaces Integral operators Low-rank approximation kernel embedding of measures Differentiable relaxation

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