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1. Data-Driven Chance Constrained Control using Kernel Distribution Embeddings

   Thorpe, Adam; and Oishi, Meeko (May 2021, Proceedings of The 4th Annual Learning for Dynamics and Control Conference)

   We present a data-driven algorithm for efficiently computing stochastic control policies for general joint chance constrained optimal control problems. Our approach leverages the theory of kernel distribution embeddings, which allows representing expectation operators as inner products in a reproducing kernel Hilbert space. This framework enables approximately reformulating the original problem using a dataset of observed trajectories from the system without imposing prior assumptions on the parameterization of the system dynamics or the structure of the uncertainty. By optimizing over a finite subset of stochastic open-loop control trajectories, we relax the original problem to a linear program over the control parameters that can be efficiently solved using standard convex optimization techniques. We demonstrate our proposed approach in simulation on a system with nonlinear non-Markovian dynamics navigating in a cluttered environment. 

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2. A Relative Spike-Timing Approach to Kernel-Based Decoding Demonstrated for Insect Flight Experiments

   https://doi.org/10.1109/IJCNN55064.2022.9892352  

   Yang, Hengye; Putney, Joy; Bin Sikandar, Usama; Zhu, Pingping; Sponberg, Simon; Ferrari, Silvia (July 2022, 2022 International Joint Conference on Neural Networks (IJCNN))

   Spike train decoding is considered one of the grand challenges in reverse-engineering neural control systems as well as in the development of neuromorphic controllers. This paper presents a novel relative-time kernel design that accounts for not only individual spike train patterns, but also the relative spike timing between neuron pairs in the population. The new relative-time-kernel-based spike train decoding method proposed in this paper allows us to map the spike trains of a population of neurons onto a lower-dimensional manifold, in which continuous-time trajectories live. The effectiveness of our novel approach is demonstrated by comparing it with existing kernel-based and rate-based decoders, including the traditional reproducing kernel Hilbert space framework. In this paper, we use the data collected in hawk moth flower tracking experiments to test the importance of relative spike timing information for neural control, and focus on the problem of uncovering the mapping from the spike trains of ten primary flight muscles to the resulting forces and torques on the moth body. We show that our new relative-time-kernel-based decoder improves the prediction of the resulting forces and torques by up to 52.1 %. Our proposed relative-time-kernel-based decoder may be used to reverse-engineer neural control systems more accurately by incorporating precise relative spike timing information in spike trains. 

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3. Banach Space Representer Theorems for Neural Networks and Ridge Splines

   Parhi, Rahul; Nowak, Robert D (February 2021, Journal of machine learning research)

   We develop a variational framework to understand the properties of the functions learned by neural networks fit to data. We propose and study a family of continuous-domain linear inverse problems with total variation-like regularization in the Radon domain subject to data fitting constraints. We derive a representer theorem showing that finite-width, singlehidden layer neural networks are solutions to these inverse problems. We draw on many techniques from variational spline theory and so we propose the notion of polynomial ridge splines, which correspond to single-hidden layer neural networks with truncated power functions as the activation function. The representer theorem is reminiscent of the classical reproducing kernel Hilbert space representer theorem, but we show that the neural network problem is posed over a non-Hilbertian Banach space. While the learning problems are posed in the continuous-domain, similar to kernel methods, the problems can be recast as finite-dimensional neural network training problems. These neural network training problems have regularizers which are related to the well-known weight decay and path-norm regularizers. Thus, our result gives insight into functional characteristics of trained neural networks and also into the design neural network regularizers. We also show that these regularizers promote neural network solutions with desirable generalization properties. Keywords: neural networks, splines, inverse problems, regularization, sparsity 

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4. On Distance and Kernel Measures of Conditional Dependence

   Sheng, Tianhong; Sriperumbudur, Bharath K. (January 2023, Journal of machine learning research)

   Measuring conditional dependence is one of the important tasks in statistical inference and is fundamental in causal discovery, feature selection, dimensionality reduction, Bayesian network learning, and others. In this work, we explore the connection between conditional dependence measures induced by distances on a metric space and reproducing kernels associated with a reproducing kernel Hilbert space (RKHS). For certain distance and kernel pairs, we show the distance-based conditional dependence measures to be equivalent to that of kernel-based measures. On the other hand, we also show that some popular kernel conditional dependence measures based on the Hilbert-Schmidt norm of a certain crossconditional covariance operator, do not have a simple distance representation, except in some limiting cases. 

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5. Functional estimation of perturbed positive real infinite dimensional systems using adaptive compensators

   https://doi.org/10.23919/ACC45564.2020.9147702  

   Demetriou, Michael A. (July 2020, 2020 American Control Conference (ACC))

   This paper extends earlier results on the adaptive estimation of nonlinear terms in finite dimensional systems utilizing a reproducing kernel Hilbert space to a class of positive real infinite dimensional systems. The simplest class of strictly positive real infinite dimensional systems has collocated input and output operators with the state operator being the generator of an exponentially stable C 0 semigroup on the state space X . The parametrization of the nonlinear term is considered in a reproducing kernel Hilbert space Q and together with the adaptive observer, results in an evolution system considered in X × Q. Using Lyapunov-redesign methods, the adaptive laws for the parameter estimates are derived and the well-posedness of the resulting evolution error system is summarized. The adaptive estimate of the unknown nonlinearity is subsequently used to compensate for the nonlinearity. A special case of finite dimensional systems with an embedded reproducing kernel Hilbert space to handle the nonlinear term is also considered and the convergence results are summarized. A numerical example on a one-dimensional diffusion equation is considered. 

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