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  • 1.
    Adler, Jonas
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). Elekta, Box 7593, 103 93 Stockholm, Sweden.
    Ringh, Axel
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Öktem, Ozan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Karlsson, Johan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Learning to solve inverse problems using Wasserstein lossManuscript (preprint) (Other academic)
    Abstract [en]

    We propose using the Wasserstein loss for training in inverse problems. In particular, we consider a learned primal-dual reconstruction scheme for ill-posed inverse problems using the Wasserstein distance as loss function in the learning. This is motivated by miss-alignments in training data, which when using standard mean squared error loss could severely degrade reconstruction quality. We prove that training with the Wasserstein loss gives a reconstruction operator that correctly compensates for miss-alignments in certain cases, whereas training with the mean squared error gives a smeared reconstruction. Moreover, we demonstrate these effects by training a reconstruction algorithm using both mean squared error and optimal transport loss for a problem in computerized tomography.

  • 2.
    Banert, Sebastian
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Ringh, Axel
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Adler, Jonas
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). Elekta, Box 7593, 103 93 Stockholm, Sweden.
    Karlsson, Johan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Öktem, Ozan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Data-driven nonsmooth optimizationManuscript (preprint) (Other academic)
  • 3.
    Karlsson, Johan
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. Shanghai Jiao Tong University, China.
    Ringh, Axel
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    The Multidimensional Moment Problem with Complexity Constraint2016In: Integral equations and operator theory, ISSN 0378-620X, E-ISSN 1420-8989, Vol. 84, no 3, p. 395-418Article in journal (Refereed)
    Abstract [en]

    A long series of previous papers have been devoted to the (one-dimensional) moment problem with nonnegative rational measure. The rationality assumption is a complexity constraint motivated by applications where a parameterization of the solution set in terms of a bounded finite number of parameters is required. In this paper we provide a complete solution of the multidimensional moment problem with a complexity constraint also allowing for solutions that require a singular measure added to the rational, absolutely continuous one. Such solutions occur on the boundary of a certain convex cone of solutions. In this paper we provide complete parameterizations of all such solutions. We also provide errata for a previous paper in this journal coauthored by one of the authors of the present paper.

  • 4.
    Karlsson, Johan
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Ringh, Axel
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Generalized Sinkhorn Iterations for Regularizing Inverse Problems Using Optimal Mass Transport2017In: SIAM Journal on Imaging Sciences, ISSN 1936-4954, E-ISSN 1936-4954, Vol. 10, no 4, p. 1935-1962Article in journal (Refereed)
    Abstract [en]

    The optimal mass transport problem gives a geometric framework for optimal allocation and has recently attracted significant interest in application areas such as signal processing, image processing, and computer vision. Even though it can be formulated as a linear programming problem, it is in many cases intractable for large problems due to the vast number of variables. A recent development addressing this builds on an approximation with an entropic barrier term and solves the resulting optimization problem using Sinkhorn iterations. In this work we extend this methodology to a class of inverse problems. In particular we show that Sinkhorn-type iterations can be used to compute the proximal operator of the transport problem for large problems. A splitting framework is then used to solve inverse problems where the optimal mass transport cost is used for incorporating a priori information. We illustrate this method on problems in computerized tomography. In particular we consider a limited-angle computerized tomography problem, where a priori information is used to compensate for missing measurements.

  • 5.
    Ringh, Axel
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Multidimensional inverse problems in imaging and identification using low-complexity models, optimal mass transport, and machine learning2018Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis, which mainly consists of six appended papers, primarily considers a number of inverse problems in imaging and system identification.

    In particular, the first two papers generalize results for the rational covariance extension problem from one to higher dimensions. The rational covariance extension problem stems from system identification and can be formulated as a trigonometric moment problem, but with a complexity constraint on the sought measure. The papers investigate a solution method based on varia tional regularization and convex optimization. We prove the existence and uniqueness of a solution to the variational problem, both when enforcing exact moment matching and when considering two different versions of approximate moment matching. A number of related questions are also considered, such as well-posedness, and the theory is illustrated with a number of examples.

    The third paper considers the maximum delay margin problem in robust control: To find the largest time delay in a feedback loop for a linear dynamical system so that there still exists a single controller that stabilizes the system for all delays smaller than or equal to this time delay. A sufficient condition for robust stabilization is recast as an analytic interpolation problem, which leads to an algorithm for computing a lower bound on the maximum delay margin. The algorithm is based on bisection, where positive semi-definiteness of a Pick matrix is used as selection criteria.

    Paper four investigate the use of optimal transport as a regularizing functional to incorporate prior information in variational formulations for image reconstruction. This is done by observing that the so-called Sinkhorn iterations, which are used to solve large scale optimal transport problems, can be seen as coordinate ascent in a dual optimization problem. Using this, we extend the idea of Sinkhorn iterations and derive a iterative algorithm for computing the proximal operator. This allows us to solve large-scale convex optimization problems that include an optimal transport term.

    In paper five, optimal transport is used as a loss function in machine learning for inverse problems in imaging. This is motivated by noise in the training data which has a geometrical characteristic. We derive theoretical results that indicate that optimal transport is better at compensating for this type of noise, compared to the standard 2-norm, and the effect is demonstrated in a numerical experiment.

    The sixth paper considers using machine learning techniques for solving large-scale convex optimization problems. We first parametrizes a family of algorithms, from which a new optimization algorithm is derived. Then we apply machine learning techniques to learn optimal parameters for given families of optimization problems, while imposing a fixed number of iterations in the scheme. By constraining the parameters appropriately, this gives learned optimization algorithms with provable convergence.

  • 6.
    Ringh, Axel
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Karlsson, Johan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. Shanghai Jiao Tong Univ, Dept Automat, Shanghai, Peoples R China; Shanghai Jiao Tong Univ, Sch Math, Shanghai, Peoples R China.
    Further results on multidimensional rational covariance extension with application to texture generation2017In: 2017 IEEE 56th Annual Conference on Decision and Control (CDC), IEEE , 2017Conference paper (Refereed)
    Abstract [en]

    The rational covariance extension problem is a moment problem with several important applications in systems and control as, for example, in identification, estimation, and signal analysis. Here we consider the multidimensional counterpart and present new results for the well-posedness of the problem. We apply the theory to texture generation by modeling the texture as the output of a Wiener system. The static nonlinearity in the Wiener system is assumed to be a thresholding function and we identify both the linear dynamical system and the thresholding parameter.

  • 7.
    Ringh, Axel
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Karlsson, Johan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. Shanghai Jiao Tong Univ, Dept Automat, Shanghai, Peoples R China; Shanghai Jiao Tong Univ, Sch Math, Shanghai, Peoples R China.
    Lower bounds on the maximum delay margin by analytic interpolation2018In: 2018 IEEE 57th Annual Conference on Decision and Control (CDC), Institute of Electrical and Electronics Engineers (IEEE), 2018, p. 5463-5469, article id 8618930Conference paper (Refereed)
    Abstract [en]

    We study the delay margin problem in the context of recent works by T. Qi, J. Zhu, and J. Chen, where a sufficient condition for the maximal delay margin is formulated in terms of an interpolation problem obtained after introducing a rational approximation. Instead we omit the approximation step and solve the same problem directly using techniques from function theory and analytic interpolation. Furthermore, we introduce a constant shift in the domain of the interpolation problem. In this way we are able to improve on their lower bound for the maximum delay margin.

  • 8.
    Ringh, Axel
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Karlsson, Johan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, Superseded Departments (pre-2005), Mathematics. Department of Automation, Shanghai Jiao Tong University, Shanghai, China.
    Multidimensional rational covariance extensionManuscript (preprint) (Other academic)
    Abstract [en]

    The rational covariance extension problem (RCEP) is an important problem in systems and control occurring in such diverse fields as control, estimation, system identification, and signal and image processing, leading to many fundamental theoretical questions. In fact, this inverse problem is a key component in many identification and signal processing techniques and plays a fundamental role in prediction, analysis, and modeling of systems and signals. It is well-known that the RCEP can be reformulated as a (truncated) trigonometric moment problem subject to a rationality condition. In this paper we consider the more general multidimensional trigonometric moment problem with a similar rationality constraint. This generalization creates many interesting new mathematical questions and also provides new insights into the original one-dimensional problem. A key concept in this approach is the complete smooth parametrization of all solutions, allowing solutions to be tuned to satisfy additional design specifications without violating the complexity constraints. As an illustration of the potential of this approach we apply our results to multidimensional spectral estimation, Wiener system identification, and image compression.

  • 9.
    Ringh, Axel
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Karlsson, Johan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. Shanghai Jiao Tong Univ, Dept Automat & Math, Shanghai 200240, Peoples R China..
    MULTIDIMENSIONAL RATIONAL COVARIANCE EXTENSION WITH APPLICATIONS TO SPECTRAL ESTIMATION AND IMAGE COMPRESSION2016In: SIAM Journal of Control and Optimization, ISSN 0363-0129, E-ISSN 1095-7138, Vol. 54, no 4, p. 1950-1982Article in journal (Refereed)
    Abstract [en]

    The rational covariance extension problem (RCEP) is an important problem in systems and control occurring in such diverse fields as control, estimation, system identification, and signal and image processing, leading to many fundamental theoretical questions. In fact, this inverse problem is a key component in many identification and signal processing techniques and plays a fundamental role in prediction, analysis, and modeling of systems and signals. It is well known that the RCEP can be reformulated as a (truncated) trigonometric moment problem subject to a rationality condition. In this paper we consider the more general multidimensional trigonometric moment problem with a similar rationality constraint. This generalization creates many interesting new mathematical questions and also provides new insights into the original one-dimensional problem. A key concept in this approach is the complete smooth parameterization of all solutions, allowing solutions to be tuned to satisfy additional design specifications without violating the complexity constraints. As an illustration of the potential of this approach we apply our results to multidimensional spectral estimation and image compression. This is just a first step in this direction, and we expect that more elaborate tuning strategies will enhance our procedures in the future.

  • 10.
    Ringh, Axel
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Karlsson, Johan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. Shanghai Jiao Tong University, China.
    Multidimensional rational covariance extension with approximate covariance matching2018In: SIAM Journal of Control and Optimization, ISSN 0363-0129, E-ISSN 1095-7138, Vol. 56, no 2, p. 913-944Article in journal (Refereed)
    Abstract [en]

    In our companion paper [A. Ringh, J. Karlsson, and A. Lindquist, SIAM T. Control Opton., 54 (2016), pp. 1950-1982] we discussed the multidimensional rational covariance extension problem (RCEP), which has important applications in image processing and spectral estimation in radar, sonar, and medical imaging. This is an inverse problem where a power spectrum with a rational absolutely continuous part is reconstructed from a finite set of moments. However, in most applications these moments are determined from observed data and are therefore only approximate, and the RCEP may not have a solution. In this paper we extend the results of our companion paper to handle approximate covariance matching. We consider two problems, one with a soft constraint and the other one with a hard constraint, and show that they are connected via a homeomorphism. We also demonstrate that the problems are well-posed and illustrate the theory by examples in spectral estimation and texture generation.

  • 11.
    Ringh, Axel
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Karlsson, Johan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. Shanghai Jiao Tong University, China.
    The Multidimensional Circulant Rational Covariance Extension Problem: Solutions and Applications in Image Compression2016In: 2015 54th IEEE Conference on Decision and Control (CDC), 2015, Institute of Electrical and Electronics Engineers (IEEE), 2016, p. 5320-5327Conference paper (Refereed)
    Abstract [en]

    Rational functions play a fundamental role in systems engineering for modelling, identification, and control applications. In this paper we extend the framework by Lindquist and Picci for obtaining such models from the circulant trigonometric moment problems, from the one-dimensional to the multidimensional setting in the sense that the spectrum domain is multidimensional. We consider solutions to weighted entropy functionals, and show that all rational solutions of certain bounded degree can be characterized by these. We also consider identification of spectra based on simultaneous covariance and cepstral matching, and apply this theory for image compression. This provides an approximation procedure for moment problems where the moment integral is over a multidimensional domain, and is also a step towards a realization theory for random fields.

  • 12.
    Ringh, Axel
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Karlsson, Johan Mikael
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    A fast solver for the circulant rational covariance extension problem2015In: 2015 European Control Conference, ECC 2015, Institute of Electrical and Electronics Engineers (IEEE), 2015, p. 727-733Conference paper (Refereed)
    Abstract [en]

    The rational covariance extension problem is to parametrize the family of rational spectra of bounded degree that matches a given set of covariances. This article treats a circulant version of this problem, where the underlying process is periodic and we seek a spectrum that also matches a set of given cepstral coefficients. The interest in the circulant problem stems partly from the fact that this problem is a natural approximation of the non-periodic problem, but is also a tool in itself for analysing periodic processes. We develop a fast Newton algorithm for computing the solution utilizing the structure of the Hessian. This is done by extending a current algorithm for Toeplitz-plus-Hankel systems to the block-Toeplitz-plus-block-Hankel case. We use this algorithm to reduce the computational complexity of the Newton search from O(n3) to O(n2), where n corresponds to the number of covariances and cepstral coefficients.

  • 13.
    Ringh, Axel
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Lindquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. Shanghai Jiao Tong University, China .
    Spectral estimation of periodic and skew periodic random signals and approximation of spectral densities2014In: Proceedings of the 33rd Chinese Control Conference, CCC 2014, 2014, p. 5322-5327Conference paper (Refereed)
    Abstract [en]

    This paper discusses extensions of the theory of rational covariance extension to periodic and skew-periodic processes. It is also shown how these methods can be used to construct fast algorithms for approximate spectral estimation of (non-periodic) processes.

  • 14.
    Ringh, Axel
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Zhuge, X.
    Palenstijn, W. J.
    Batenburg, K. J.
    Öktem, Ozan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    High-level algorithm prototyping: An example extending the TVR-DART algorithm2017In: Discrete Geometry for Computer Imagery: 20th IAPR International Conference, DGCI 2017, Vienna, Austria, September 19 – 21, 2017, Proceedings, Springer, 2017, p. 109-121Chapter in book (Refereed)
    Abstract [en]

    Operator Discretization Library (ODL) is an open-source Python library for prototyping reconstruction methods for inverse problems, and ASTRA is a high-performance Matlab/Python toolbox for large-scale tomographic reconstruction. The paper demonstrates the feasibility of combining ODL with ASTRA to prototype complex reconstruction methods for discrete tomography. As a case in point, we consider the total-variation regularized discrete algebraic reconstruction technique (TVR-DART). TVR-DART assumes that the object to be imaged consists of a limited number of distinct materials. The ODL/ASTRA implementation of this algorithm makes use of standardized building blocks, that can be combined in a plug-and-play manner. Thus, this implementation of TVR-DART can easily be adapted to account for application specific aspects, such as various noise statistics that come with different imaging modalities.

  • 15.
    Zhang, Silun
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Ringh, Axel
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Hu, Xiaoming
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Karlsson, Johan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    A moment-based approach to modeling collective behaviors2018In: 2018 IEEE Conference on Decision and Control (CDC), Institute of Electrical and Electronics Engineers (IEEE), 2018, p. 1681-1687, article id 8619389Conference paper (Refereed)
    Abstract [en]

    In this work we introduce an approach for modeling and analyzing collective behavior of a group of agents using moments. We represent the occupation measure of the group of agents by their moments and show how the dynamics of the moments can be modeled. Then approximate trajectories of the moments can be computed and an inverse problem is solved to recover macro-scale properties of the group of agents. To illustrate the theory, a numerical example with interactions between the agents is given.

  • 16.
    Zhang, Silun
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Ringh, Axel
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Hu, Xiaoming
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Karlsson, Johan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM.
    Modeling collective behaviors: A moment-based approachManuscript (preprint) (Other academic)
    Abstract [en]

    Abstract—In this work we introduce an approach for modeling and analyzing collective behavior of a group of agents using moments. We represent the group of agents via their distribution and derive a method to estimate the dynamics of the moments. We use this to predict the evolution of the distribution of agents by first computing the moment trajectories and then use this to reconstruct the distribution of the agents. In the latter an inverse problem is solved in order to reconstruct a nominal distribution and to recover the macro-scale properties of the group of agents. The proposed method is applicable for several types of multi-agent systems, e.g., leader-follower systems. We derive error bounds for the moment trajectories and describe how to take these error bounds into account for computing the moment dynamics. The convergence of the moment dynamics is also analyzed for cases with monomial moments. To illustrate the theory, two numerical examples are given. In the first we consider a multi-agent system with interactions and compare the proposed methods for several types of moments. In the second example we apply the framework to a leader-follower problem for modeling pedestrian crowd dynamics.

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