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Learning Approximate Forward Reachable Sets Using Separating Kernels
TL;DR: A data-driven method for computing approximate forward reachable sets using separating kernels in a reproducing kernel Hilbert space using a computationally efficient representation for the classifier that is the solution to a regularized least squares problem.
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Abstract: 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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Citations
•Posted Content
Data-Driven Stochastic Reachability Using Hilbert Space Embeddings.
TL;DR: By embedding the stochastic kernel of a Markov control process in a reproducing kernel Hilbert space, this work can compute the safety probabilities for systems with arbitrary disturbances as simple matrix operations and inner products.
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SOCKS: A Stochastic Optimal Control and Reachability Toolbox Using Kernel Methods
Adam Thorpe,Meeko M. K. Oishi +1 more
- 12 Mar 2022
TL;DR: SOCKS as mentioned in this paper is a data-driven stochastic optimal control toolbox based on kernel methods, a nonparametric technique which can be used to represent probability distributions in a high-dimensional space of functions known as reproducing kernel Hilbert space.
Fully-Automated Verification of Linear Systems Using Inner- and Outer-Approximations of Reachable Sets
TL;DR: A fully-automated reachability algorithm that tunes all algorithm parameters internally such that the reachable set enclosure has a user-defined accuracy in terms of distance to the exact reachability set.
Exact Characterization of the Convex Hulls of Reachable Sets
TL;DR: In this article , the convex hulls of reachable sets of nonlinear systems with bounded disturbances are characterized as convex solutions of an ordinary differential equation from all possible initial values of the disturbances.
Convex and Nonconvex Sublinear Regression with Application to Data-driven Learning of Reach Sets
Shadi Haddad,Abhishek Halder +1 more
- 04 Oct 2022
TL;DR: This work considers estimating a compact set fromite data by approximating the support function of that set via sublinear regression, and proposes two algorithms to perform the sub linear regression, one via convex and another via nonconvex programming.
References
Theory of Reproducing Kernels.
TL;DR: In this paper, a short historical introduction is given to indicate the different manners in which these kernels have been used by various investigators and discuss the more important trends of the application of these kernels without attempting, however, a complete bibliography of the subject matter.
•Proceedings Article
Random Features for Large-Scale Kernel Machines
Ali Rahimi,Benjamin Recht +1 more
- 03 Dec 2007
TL;DR: Two sets of random features are explored, provided convergence bounds on their ability to approximate various radial basis kernels, and it is shown that in large-scale classification and regression tasks linear machine learning algorithms applied to these features outperform state-of-the-art large- scale kernel machines.
Optimal Rates for the Regularized Least-Squares Algorithm
Andrea Caponnetto,E. De Vito +1 more
TL;DR: A complete minimax analysis of the problem is described, showing that the convergence rates obtained by regularized least-squares estimators are indeed optimal over a suitable class of priors defined by the considered kernel.
A Proposal on Machine Learning via Dynamical Systems
TL;DR: The idea of using continuous dynamical systems to model general high-dimensional nonlinear functions used in machine learning and the connection with deep learning is discussed.
912
Flow*: an analyzer for non-linear hybrid systems
Xin Chen,Erika Ábrahám,Sriram Sankaranarayanan +2 more
- 13 Jul 2013
TL;DR: The tool Flow* performs Taylor model-based flowpipe construction for non-linear (polynomial) hybrid systems with a wide variety of optimizations including adaptive step sizes, adaptive selection of approximation orders and the heuristic selection of template directions for aggregation flowpipes.