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LOCALISATION OF REGULARISED AND MULTIVIEW SUPPORT VECTOR MACHINE LEARNING

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arXiv 2023年

作者： Gheondea, Aurelian Tilki, Cankat Institutul de Matematica Academiei Române 21 Calea Griviţei Bucureşti010702 Romania Department of Mathematics Bilkent University Bilkent Ankara06800 Turkey Department of Mathematics Division of Computational Modeling and Data Analytics Virginia Polytechnic Institute and State University BlacksburgVA24061 United States

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 a particular different types of 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 *** Codes Primary 68T05, Secondary 46E22, 46G05 Copyright © 2023, The Authors. All rights reserved.

关键词： Adversarial machine learning

computational methods for large-scale inverse problems: A survey on hybrid projection methods

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arXiv 2021年

作者： Chung, Julianne Gazzola, Silvia Department of Mathematics Computational Modeling and Data Analytics Division Academy of Integrated Science Virginia Tech BlacksburgVA United States Department of Mathematical Sciences University of Bath United Kingdom

This paper surveys an important class of methods that combine iterative projection methods and variational regularization methods for large-scale inverse problems. Iterative methods such as Krylov subspace methods are invaluable in the numerical linear algebra community and have proved important in solving inverse problems due to their inherent regularizing properties and their ability to handle large-scale problems. Variational regularization describes a broad and important class of methods that are used to obtain reliable solutions to inverse problems, whereby one solves a modified problem that incorporates prior knowledge. Hybrid projection methods combine iterative projection methods with variational regularization techniques in a synergistic way, providing researchers with a powerful computational framework for solving very large inverse problems. Although the idea of a hybrid Krylov method for linear inverse problems goes back to the 1980s, several recent advances on new regularization frameworks and methodologies have made this field ripe for extensions, further analyses, and new applications. In this paper, we provide a practical and accessible introduction to hybrid projection methods in the context of solving large (linear) inverse problems. Copyright © 2021, The Authors. All rights reserved.

关键词： Differential equations

Structured interpolation for multivariate transfer functions of quadratic-bilinear systems

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arXiv 2023年

作者： Benner, Peter Gugercin, Serkan Werner, Steffen W.R. Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstraße 1 Magdeburg39106 Germany Otto von Guericke University Magdeburg Faculty of Mathematics Universitätsplatz 2 Magdeburg39106 Germany Department of Mathematics and 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

High-dimensional/high-fidelity nonlinear dynamical systems appear naturally when the goal is to accurately model real-world phenomena. Many physical properties are thereby encoded in the internal differential structure of these resulting large-scale nonlinear systems. The high-dimensionality of the dynamics causes computational bottlenecks, especially when these large-scale systems need to be simulated for a variety of situations such as different forcing terms. This motivates model reduction where the goal is to replace the full-order dynamics with accurate reduced-order surrogates. Interpolation-based model reduction has been proven to be an effective tool for the construction of cheap-to-evaluate surrogate models that preserve the internal structure in the case of weak nonlinearities. In this paper, we consider the construction of multivariate interpolants in frequency domain for structured quadratic-bilinear systems. We propose definitions for structured variants of the symmetric subsystem and generalized transfer functions of quadratic-bilinear systems and provide conditions for structure-preserving interpolation by projection. The theoretical results are illustrated using two numerical examples including the simulation of molecular dynamics in crystal *** Codes 30E05, 34K17, 65D05, 93C10, 93A15 Copyright © 2023, The Authors. All rights reserved.

关键词： Interpolation

Hybrid Projection Methods for Solution Decomposition in Large-scale Bayesian Inverse Problems

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arXiv 2022年

作者： Chung, Julianne Jiang, Jiahua Miller, Scot M. Saibaba, Arvind K. Department of Mathematics Computational Modeling and Data Analytics Division Academy of Integrated Science Virginia Tech BlacksburgVA United States Department of Mathematics University of Birmingham United Kingdom Department of Environmental Health and Engineering Johns Hopkins University United States Department of Mathematics North Carolina State University United States

We develop hybrid projection methods for computing solutions to large-scale inverse problems, where the solution represents a sum of different stochastic components. Such scenarios arise in many imaging applications (e.g., anomaly detection in atmospheric emissions tomography) where the reconstructed solution can be represented as a combination of two or more components and each component contains different smoothness or stochastic properties. In a deterministic inversion or inverse modeling framework, these assumptions correspond to different regularization terms for each solution in the sum. Although various prior assumptions can be included in our framework, we focus on the scenario where the solution is a sum of a sparse solution and a smooth solution. For computing solution estimates, we develop hybrid projection methods for solution decomposition that are based on a combined flexible and generalized Golub-Kahan processes. This approach integrates techniques from the generalized Golub-Kahan bidiagonalization and the flexible Krylov methods. The benefits of the proposed methods are that the decomposition of the solution can be done iteratively, and the regularization terms and regularization parameters are adaptively chosen at each iteration. Numerical results from photoacoustic tomography and atmospheric inverse modeling demonstrate the potential for these methods to be used for anomaly detection. Copyright © 2022, The Authors. All rights reserved.

关键词： Inverse problems

Structure-preserving interpolation of quadratic-bilinear systems via regular multivariate transfer functions

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PAMM 2024年第3期24卷

作者： Peter Benner Serkan Gugercin Steffen W. R. Werner Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg Germany Faculty of Mathematics Otto von Guericke University Magdeburg Magdeburg Germany Division of Computational Modeling and Data Analytics Department of Mathematics Academy of Data Science Virginia Tech Blacksburg Virginia USA

For the accurate modeling of real-world phenomena, high-dimensional nonlinear dynamical systems are indispensable. Quadratic-bilinear nonlinearities are of particular importance since many systems with general nonlinearities can be remodeled into this form. Such nonlinear systems typically encode physical properties via their internal differential structure. This leads to a high demand for structure-preserving model order reduction techniques for quadratic-bilinear systems that allow for the accurate representation of their dynamics by a small number of differential equations while retaining the internal system structure. In this paper, we provide formulae for structured regular subsystem transfer functions of quadratic-bilinear systems with internal differential structures and provide a corresponding interpolation theory for the efficient construction of the resulting structured reduced-order models. The presented theoretical results are illustrated in numerical experiments including a time-delayed nonlinear reaction–diffusion equation and a molecular simulation.

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data-driven balancing of linear dynamical systems

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arXiv 2021年

作者： Gosea, Ion Victor Gugercin, Serkan Beattie, Christopher Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany Department of Mathematics Computational Modeling and Data Analytics Division Academy of Integrated Science Virginia Tech BlacksburgVA24061 United States

We present a novel reformulation of balanced truncation, a classical model reduction method. The principal innovation that we introduce comes through the use of system response data that has been either measured or computed, without reference to any prescribed realization of the original model. data are represented by sampled values of the transfer function or the impulse response corresponding to the original model. We discuss parallels that our approach bears with the Loewner framework, another popular data-driven method. We illustrate our approach numerically in both continuous-time and discrete-time *** Codes 37M99, 41A20, 65D30, 65K99, 93A15, 93B15 Copyright © 2021, The Authors. All rights reserved.

关键词： Impulse response

data-driven modeling of linear dynamical systems with quadratic output in the AAA framework

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arXiv 2021年

作者： Gosea, Ion Victor Gugercin, Serkan Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany Department of Mathematics and Computational Modeling and Data Analytics Division Academy of Integrated Science Virginia Tech BlacksburgVA24061 United States

We extend the AAA (Adaptive-Antoulas-Anderson) algorithm to develop a data-driven modeling framework for linear systems with quadratic output (LQO). Such systems are characterized by two transfer functions: one corresponding to the linear part of the output and another one to the quadratic part. We first establish the joint barycentric representations and the interpolation theory for the two transfer functions of LQO systems. This analysis leads to the proposed AAA-LQO algorithm. We show that by interpolating the transfer function values on a subset of samples together with imposing a least-squares minimization on the rest, we construct reliable data-driven LQO models. Two numerical test cases illustrate the efficiency of the proposed *** Codes 41A20, 93D25, 93C10 Copyright © 2021, The Authors. All rights reserved.

关键词： Interpolation

A unifying framework for tangential interpolation of structured bilinear control systems

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arXiv 2022年

作者： Benner, Peter Gugercin, Serkan Werner, Steffen W.R. Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstraße 1 Magdeburg39106 Germany Otto von Guericke University Magdeburg Faculty of Mathematics Universitätsplatz 2 Magdeburg39106 Germany 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

In this paper, we consider the structure-preserving model order reduction problem for multi-input/multi-output bilinear control systems by tangential interpolation. We propose a new type of tangential interpolation problem for structured bilinear systems, for which we develop a new structure-preserving interpolation framework. This new framework extends and generalizes different formulations of tangential interpolation for bilinear systems from the literature and also provides a unifying framework. We then derive explicit conditions on the projection spaces to enforce tangential interpolation in different settings, including conditions for tangential Hermite interpolation. The analysis is illustrated by means of three numerical *** Codes 30E05, 34K17, 65D05, 93A15, 93C10 Copyright © 2022, The Authors. All rights reserved.

关键词： Interpolation

Hybrid projection methods with recycling for inverse problems

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arXiv 2020年

作者： Chung, Julianne Sturler, Eric D.E. Jiang, Jiahua Department of Mathematics Computational Modeling and Data Analytics Division Academy of Integrated Science Virginia Tech BlacksburgVA United States Department of Mathematics Computational Modeling and Data Analytics Division Academy of Integrated Science Virginia Tech BlacksburgVA United States Department of Mathematics Virginia Tech BlacksburgVA United States

Iterative hybrid projection methods have proven to be very effective for solving large linear inverse problems due to their inherent regularizing properties as well as the added flexibility to select regularization parameters adaptively. In this work, we develop Golub-Kahan-based hybrid projection methods that can exploit compression and recycling techniques in order to solve a broad class of inverse problems where memory requirements or high computational cost may otherwise be prohibitive. For problems that have many unknown parameters and require many iterations, hybrid projection methods with recycling can be used to compress and recycle the solution basis vectors to reduce the number of solution basis vectors that must be stored, while obtaining a solution accuracy that is comparable to that of standard methods. If reorthogonalization is required, this may also reduce computational cost substantially. In other scenarios, such as streaming data problems or inverse problems with multiple datasets, hybrid projection methods with recycling can be used to efficiently integrate previously computed information for faster and better reconstruction. Additional benefits of the proposed methods are that various subspace selection and compression techniques can be incorporated, standard techniques for automatic regularization parameter selection can be used, and the methods can be applied multiple times in an iterative fashion. Theoretical results show that, under reasonable conditions, regularized solutions for our proposed recycling hybrid method remain close to regularized solutions for standard hybrid methods and reveal important connections among the resulting projection matrices. Numerical examples from image processing show the potential benefits of combining recycling with hybrid projection methods. Copyright © 2020, The Authors. All rights reserved.

关键词： Inverse problems