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Localisation of regularised and multiview support vector machine learning

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The Journal of Machine Learning Research

The Journal of Machine Learning Research 2024年第1期25卷 18029-18075页

作者： Aurelian Gheondea Cankat Tilki Institute of Mathematics of the Romanian Academy Bucharest Romania and Department of Mathematics Bilkent University Bilkent Ankara Turkey Department of Mathematics and Division of Computational Modeling and Data Analytics Virginia Polytechnic Institute and State University Blacksburg Virginia

We prove some representer theorems for a localised version of a semisupervised, manifold regularised and multiview support vector machine learning problem introduced by H.Q. Minh, L. Bazzani, and V. Murino, Journal of Machine Learning Research, 17(2016) 1-72, that involves operator valued positive semidefinite kernels and their reproducing kernel Hilbert spaces. The results concern general cases when convex or nonconvex loss functions and finite or infinite dimensional underlying Hilbert spaces are considered. We show that the general framework allows infinite dimensional Hilbert spaces and nonconvex loss functions for some special cases, in particular in case the loss functions are Gâteaux differentiable. Detailed calculations are provided for the exponential least squares loss functions that lead to systems of partially nonlinear equations for which some Newton's approximation methods based on the interior point method can be used. Some numerical experiments are performed on a toy model that illustrate the tractability of the methods that we propose.

关键词： operator valued reproducing kernel Hilbert spaces manifold co-regularised and multiview learning support vector machine learning loss functions representer theorem

Interpolatory model reduction of dynamical systems with root mean squared error

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arXiv 2024年

作者： Reiter, Sean Werner, Steffen W.R. Department of Mathematics Virginia Tech BlacksburgVA24061 United States Department of Mathematics Division of Computational Modeling and Data Analytics Academy of Data Science Virginia Tech BlacksburgVA24061 United States

The root mean squared error is an important measure used in a variety of applications such as structural dynamics and acoustics to model averaged deviations from standard behavior. For large-scale systems, simulations of this quantity quickly become computationally prohibitive. Classical model order reduction techniques attempt to resolve this issue via the construction of surrogate models that emulate the root mean squared error measure using an intermediate linear system. However, this approach requires a potentially large number of linear outputs, which can be disadvantageous in the design of reduced-order models. In this work, we consider directly the root mean squared error as the quantity of interest using the concept of quadratic-output models and propose several new model reduction techniques for the construction of appropriate surrogates. We test the proposed methods on a model for the vibrational response of a plate with tuned vibration absorbers. Copyright © 2024, The Authors. All rights reserved.

关键词： Mean square error

TIME-DOMAIN ITERATIVE RATIONAL KRYLOV METHOD

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arXiv 2024年

作者： Ackermann, Michael S. Gugercin, Serkan Department of Mathematics Virginia Tech BlacksburgVA24061 United States Department of Mathematics Division of Computational Modeling and Data Analytics Academy of Data Science Virginia Tech BlacksburgVA24061 United States

The Realization Independent Iterative Rational Krylov Algorithm (TF-IRKA) is a frequency-based data-driven reduced order modeling (DDROM) method that constructs H2 optimal DDROMs. However, as the H2 optimal approximation theory dictates, TF-IRKA requires repeated sampling of frequency data, that is, values of the system transfer function and its derivative, outside the unit circle. This repeated evaluation of frequency data requires repeated full model computations and may not be feasible. The data-informativity framework for moment matching provides a method for obtaining such frequency data from a single time-domain simulation. However, this framework usually requires solving linear systems with prohibitively ill-conditioned matrices, especially when recovering frequency data from off the unit circle as required for optimality. In this paper, building upon our previous work with the data informativity framework for moment matching, we provide a formula for the nonzero extreme eigenvalues of a symmetric rank-1 perturbation to an orthogonal projection, which then leads to an optimal scaling of the aforementioned linear systems. We also establish connections between the underlying dynamical system and conditioning of these linear systems. This analysis then leads to our algorithmic development, time-domain IRKA, which allows us to implement a time-domain variant of TF-IRKA, constructing H2 optimal DDROMs from a single time-domain simulation without requiring repeated frequency data evaluations. The numerical examples illustrate the effectiveness of the proposed algorithm. Copyright © 2024, The Authors. All rights reserved.

关键词： Linear systems

System stabilization with policy optimization on unstable latent manifolds

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arXiv 2024年

作者： Werner, Steffen W.R. Peherstorfer, Benjamin Department of Mathematics Division of Computational Modeling and Data Analytics Academy of Data Science Virginia Tech BlacksburgVA24061 United States Courant Institute of Mathematical Sciences New York University New YorkNY10012 United States

Stability is a basic requirement when studying the behavior of dynamical systems. However, stabilizing dynamical systems via reinforcement learning is challenging because only little data can be collected over short time horizons before instabilities are triggered and data become meaningless. This work introduces a reinforcement learning approach that is formulated over latent manifolds of unstable dynamics so that stabilizing policies can be trained from few data samples. The unstable manifolds are minimal in the sense that they contain the lowest dimensional dynamics that are necessary for learning policies that guarantee stabilization. This is in stark contrast to generic latent manifolds that aim to approximate all—stable and unstable—system dynamics and thus are higher dimensional and often require higher amounts of data. Experiments demonstrate that the proposed approach stabilizes even complex physical systems from few data samples for which other methods that operate either directly in the system state space or on generic latent manifolds *** Codes 37N35, 68T07, 90C30, 93C57, 93D15 Copyright © 2024, The Authors. All rights reserved.

关键词： Dynamical systems

Empirical sparse regression on quadratic manifolds-

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arXiv 2024年

作者： Schwerdtner, Paul Gugercin, Serkan Peherstorfer, Benjamin Courant Institute of Mathematical Sciences New York University New YorkNY10012 United States Department of Mathematics Division of Computational Modeling and Data Analytics Academy of Data Science Virginia Tech BlacksburgVA24061 United States

Approximating field variables and data vectors from sparse samples is a key challenge in computational science. Widely used methods such as gappy proper orthogonal decomposition and empirical interpolation rely on linear approximation spaces, limiting their effectiveness for data representing transport-dominated and wave-like dynamics. To address this limitation, we introduce quadratic manifold sparse regression, which trains quadratic manifolds with a sparse greedy method and computes approximations on the manifold through novel nonlinear projections of sparse samples. The nonlinear approximations obtained with quadratic manifold sparse regression achieve orders of magnitude higher accuracies than linear methods on data describing transport-dominated dynamics in numerical *** Codes 65F55, 62H25, 65F30, 68T09, 65F20, 65M22 Copyright © 2024, The Authors. All rights reserved.

关键词： Nonlinear simulations

Using LDLT factorizations in Newton's method for solving general large-scale algebraic Riccati equations

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arXiv 2024年

作者： Saak, Jens Werner, Steffen W.R. Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstraße 1 Magdeburg39106 Germany Department of Mathematics Division of Computational Modeling and Data Analytics Academy of Data Science Virginia Tech BlacksburgVA24061 United States

Continuous-time algebraic Riccati equations can be found in many disciplines in different forms. In the case of small-scale dense coefficient matrices, stabilizing solutions can be computed to all possible formulations of the Riccati equation. This is not the case when it comes to large-scale sparse coefficient matrices. In this paper, we provide a reformulation of the Newton-Kleinman iteration scheme for continuous-time algebraic Riccati equations using indefinite symmetric low-rank factorizations. This allows the application of the method to the case of general large-scale sparse coefficient matrices. We provide convergence results for several prominent realizations of the equation and show in numerical examples the effectiveness of the *** Codes 15A24, 49M15, 65F45, 65H10, 93A15 Copyright © 2024, The Authors. All rights reserved.

关键词： Continuous time systems

FREQUENCY-BASED REDUCED MODELS FROM PURELY TIME-DOMAIN data VIA data INFORMATIVITY

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arXiv 2023年

Frequency-based methods have been successfully employed in creating high-fidelity data-driven reduced order models (DDROMs) for linear dynamical systems. These methods require access to values (and sometimes derivatives) of the frequency-response function (transfer function) in the complex plane. These frequency domain values can at times be costly or difficult to obtain (especially if the method of choice requires resampling);instead one may have access to only time-domain input-output data. The data informativity approach to moment matching provides a powerful new framework for recovering the required frequency data from a single time-domain trajectory. In this work, we analyze and extend upon this framework, resulting in significantly improved conditioning of the associated linear systems, an error indicator, and the removal of an assumption that the system order is known. This analysis leads to a robust algorithm for recovering frequency information from time-domain data, suitable for large-scale systems. We demonstrate the effectiveness of our algorithm by forming frequency-based DDROMs from time-domain data of several dynamical systems. Copyright © 2023, The Authors. All rights reserved.

关键词： Transfer functions

Balanced truncation with conformal maps

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arXiv 2024年

作者： Borghi, Alessandro Breiten, Tobias Gugercin, Serkan Technical University of Berlin Mathematics Department Straße des 17. Juni 136 Berlin10623 Germany Department of Mathematics Division of Computational Modeling and Data Analytics Academy of Data Science Virginia Tech BlacksburgVA24061 United States

We consider the problem of constructing reduced models for large scale systems with poles in general domains in the complex plane (as opposed to, e.g., the open left-half plane or the open unit disk). Our goal is to design a model reduction scheme, building upon theoretically established methodologies, yet encompassing this new class of models. To this aim, we develop a balanced truncation framework through conformal maps to handle poles in general domains. The major difference from classical balanced truncation resides in the formulation of the Gramians. We show that these new Gramians can still be computed by solving modified Lyapunov equations for specific conformal maps. A numerical algorithm to perform balanced truncation with conformal maps is developed and is tested on three numerical examples, namely a heat model, the Schrödinger equation, and the undamped linear wave equation, the latter two having spectra on the imaginary axis. Copyright © 2024, The Authors. All rights reserved.

关键词： Poles

KLAP: KYP LEMMA BASED LOW-RANK APPROXIMATION FOR H2-OPTIMAL PASSIVATION

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arXiv 2025年

作者： Nicodemus, Jonas Voigt, Matthias Gugercin, Serkan Unger, Benjamin University of Stuttgart Universitätsstr. 32 Stuttgart70569 Germany Department of Mathematics Division of Computational Modeling and Data Analytics Academy of Data Science Virginia Tech BlacksburgVA24061 United States Faculty of Mathematics and Computer Science UniDistance Suisse Schinerstr. 18 Brig3900 Switzerland

We present a novel passivity enforcement (passivation) method, called KLAP, for linear time-invariant systems based on the Kalman-Yakubovich-Popov (KYP) lemma and the closely related Lur’e equations. The passivation problem in our framework corresponds to finding a perturbation to a given non-passive system that renders the system passive while minimizing the H2 or frequency-weighted H2 distance between the original non-passive and the resulting passive system. We show that this problem can be formulated as an unconstrained optimization problem whose objective function can be differentiated efficiently even in large-scale settings. We show that any minimizer of the unconstrained problem yields the same passive system. Furthermore, we prove that, in the absence of a feedthrough term, every local minimizer is also a global minimizer. For cases involving a non-trivial feedthrough term, we analyze global minimizers in relation to the extremal solutions of the Lur’e equations, which can serve as tools for identifying local minima. To solve the resulting numerical optimization problem efficiently, we propose an initialization strategy based on modifying the feedthrough term and a restart strategy when it is likely that the optimization has converged to a local minimum. Numerical examples illustrate the effectiveness of the proposed *** Codes 37J06, 37M99, 65P10, 93A30, 93B30, 93C05 © 2025, CC BY.

关键词： Convergence of numerical methods