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1. Adaptive RKHS-based functional estimation of structurally perturbed second order infinite dimensional systems

   https://doi.org/10.1109/CDC51059.2022.9992577  

   Demetriou, Michael A. (December 2022, 2022 IEEE 61st Conference on Decision and Control (CDC))

   . (Ed.)

   This paper proposes a new approach for the adaptive functional estimation of second order infinite dimensional systems with structured perturbations. First, the proposed observer is formulated in the natural second order setting thus ensuring the time derivative of the estimated position is the estimated velocity, and therefore called natural adaptive observer. Assuming that the system does not yield a positive real system when placed in first order form, then the next step in deriving parameter adaptive laws is to assume a form of input-output collocation. Finally, to estimate structured perturbations taking the form of functions of the position and/or velocity outputs, the parameter space is not identified by a finite dimensional Euclidean space but instead is considered in a Reproducing Kernel Hilbert Space. Such a setting allows one not to be restricted by a priori assumptions on the dimension of the parameter spaces. Convergence of the position and velocity errors in their respective norms is established via the use of a parameter-dependent Lyapunov function, specifically formulated for second order infinite dimensional systems that include appropriately defined norms of the functional errors in the reproducing kernel Hilbert spaces. Boundedness of the functional estimates immediately follow and via an appropriate definition of a persistence of excitation condition for functional estimation, a functional convergence follows. When the system is governed by vector second order dynamics, all abstract spaces for the state evolution collapse to a Euclidean space and the natural adaptive observer results simplify. Numerical results of a second order PDE and a multi-degree of freedom finite dimensional mechanical system are presented. 

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2. Fault detection and accommodation of positive real infinite dimensional systems using adaptive RKHS-based functional estimation

   https://doi.org/10.23919/ACC53348.2022.9867405  

   Demetriou, Michael A. (June 2022, 2022 American Control Conference)

   This paper presents an adaptive functional estimation scheme for the fault detection and diagnosis of nonlinear faults in positive real infinite dimensional systems. The system is assumed to satisfy a positive realness condition and the fault, taking the form of a nonlinear function of the output, is assumed to enter the system at an unknown time. The proposed detection and diagnostic observer utilizes a Reproducing Kernel Hilbert Space as the parameter space and via a Lyapunov redesign approach, the learning scheme for the unknown functional is used for the detection of the fault occurrence, the diagnosis of the fault and finally its accommodation via an adaptive control reconfiguration. Results on parabolic PDEs with either boundary or in-domain actuation and sensing are included. 

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3. Pure Exploration in Kernel and Neural Bandits

   Zhu, Yinglun; Zhou, Dongruo; Jiang, Ruoxi; Gu, Quanquan; Willett, Rebecca; Nowak, Robert D (November 2021, Advances in Neural Information Processing Systems)

   We study pure exploration in bandits, where the dimension of the feature representation can be much larger than the number of arms. To overcome the curse of dimensionality, we propose to adaptively embed the feature representation of each arm into a lower-dimensional space and carefully deal with the induced model misspecifications. Our approach is conceptually very different from existing works that can either only handle low-dimensional linear bandits or passively deal with model misspecifications. We showcase the application of our approach to two pure exploration settings that were previously under-studied: (1) the reward function belongs to a possibly infinite-dimensional Reproducing Kernel Hilbert Space, and (2) the reward function is nonlinear and can be approximated by neural networks. Our main results provide sample complexity guarantees that only depend on the effective dimension of the feature spaces in the kernel or neural representations. Extensive experiments conducted on both synthetic and real-world datasets demonstrate the efficacy of our methods. 

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4. Two-sample statistics based on anisotropic kernels

   https://doi.org/10.1093/imaiai/iaz018  

   Cheng, Xiuyuan; Cloninger, Alexander; Coifman, Ronald R. (December 2019, Information and Inference: A Journal of the IMA)

   Abstract The paper introduces a new kernel-based Maximum Mean Discrepancy (MMD) statistic for measuring the distance between two distributions given finitely many multivariate samples. When the distributions are locally low-dimensional, the proposed test can be made more powerful to distinguish certain alternatives by incorporating local covariance matrices and constructing an anisotropic kernel. The kernel matrix is asymmetric; it computes the affinity between $$n$$ data points and a set of $$n_R$$ reference points, where $$n_R$$ can be drastically smaller than $$n$$. While the proposed statistic can be viewed as a special class of Reproducing Kernel Hilbert Space MMD, the consistency of the test is proved, under mild assumptions of the kernel, as long as $$\|p-q\| \sqrt{n} \to \infty $$, and a finite-sample lower bound of the testing power is obtained. Applications to flow cytometry and diffusion MRI datasets are demonstrated, which motivate the proposed approach to compare distributions. 

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5. Learning Approximate Forward Reachable Sets Using Separating Kernels

   Thorpe, Adam; Ortiz, Kendric; and Oishi, Meeko M. (January 2021, Proceedings of Machine Learning Research)

   null (Ed.)

   We present a data-driven method for computing approximate forward reachable sets using separating kernels in a reproducing kernel Hilbert space. We frame the problem as a support estimation problem, and learn a classifier of the support as an element in a reproducing kernel Hilbert space using a data-driven approach. Kernel methods provide a computationally efficient representation for the classifier that is the solution to a regularized least squares problem. The solution converges almost surely as the sample size increases, and admits known finite sample bounds. This approach is applicable to stochastic systems with arbitrary disturbances and neural network verification problems by treating the network as a dynamical system, or by considering neural network controllers as part of a closed-loop system. We present our technique on several examples, including a spacecraft rendezvous and docking problem, and two nonlinear system benchmarks with neural network controllers. 

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