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  • 1.
    Blomberg, Niclas
    KTH, School of Electrical Engineering (EES), Automatic Control.
    On Nuclear Norm Minimization in System Identification2016Licentiate thesis, monograph (Other academic)
    Abstract [en]

    In system identification we model dynamical systems from measured data. This data-driven approach to modelling is useful since many real-world systems are difficult to model with physical principles. Hence, a need for system identification arises in many applications involving simulation, prediction, and model-based control.

    Some of the classical approaches to system identification can lead to numerically intractable or ill-posed optimization problems. As an alternative, it has recently been shown beneficial to use so called regularization techniques, which make the ill-posed problems ‘regular’. One type of regularization is to introduce a certain rank constraint. However, this in general still leads to a numerically intractable problem, since the rank function is non-convex. One possibility is then use a convex approximation of rank, which we will do here.

    The nuclear norm, i.e., the sum of the singular values, is a popular, convex surrogate of the rank function. This results in a heuristic that has been widely used in e.g. signal processing, machine learning, control, and system identification, since its introduction in 2001. The nuclear norm heuristic introduces a regularization parameter which governs the trade-off between model fit and model complexity. The parameter is difficult to tune, and the

    current thesis revolves around this issue.

    In this thesis, we first propose a choice of the regularization parameter based on the statistical properties of fictitious validation data. This can be used to avoid computationally costly techniques such as cross-validation, where the problem is solved multiple times to find a suitable parameter value. The proposed choice can also be used as initialization to search methods for minimizing some criterion, e.g. a validation cost, over the parameter domain.

    Secondly, we study how the estimated system changes as a function of the parameter over its entire domain, which can be interpreted as a sensitivity analysis. For this we suggest an algorithm to compute a so called approximate regularization path with error guarantees, where the regularization path is the optimal solution as a function of the parameter. We are then able to guarantee the model fit, or, alternatively, the nuclear norm of the approximation, to deviate from the optimum by less than a pre-specified tolerance. Furthermore, we bound the l2-norm of the Hankel singular value approximation error, which means that in a certain subset of the parameter domain, we can guarantee the optimal Hankel singular values returned by the nuclear norm heuristic to not change more (in l2-norm) than a bounded, known quantity.

    Our contributions are demonstrated and evaluated by numerical examples using simulated and benchmark data.

  • 2.
    Blomberg, Niclas
    et al.
    KTH, School of Electrical Engineering (EES), Automatic Control.
    Rojas, Cristian R.
    KTH, School of Electrical Engineering (EES), Automatic Control.
    Wahlberg, Bo
    KTH, School of Electrical Engineering (EES), Automatic Control.
    Approximate regularization path for nuclear norm based H2 model reduction2014In: Proceedings of the IEEE Conference on Decision and Control, IEEE conference proceedings, 2014, no February, p. 3637-3641Conference paper (Refereed)
    Abstract [en]

    This paper concerns model reduction of dynamical systems using the nuclear norm of the Hankel matrix to make a trade-off between model fit and model complexity. This results in a convex optimization problem where this tradeoff is determined by one crucial design parameter. The main contribution is a methodology to approximately calculate all solutions up to a certain tolerance to the model reduction problem as a function of the design parameter. This is called the regularization path in sparse estimation and is a very important tool in order to find the appropriate balance between fit and complexity. We extend this to the more complicated nuclear norm case. The key idea is to determine when to exactly calculate the optimal solution using an upper bound based on the so-called duality gap. Hence, by solving a fixed number of optimization problems the whole regularization path up to a given tolerance can be efficiently computed. We illustrate this approach on some numerical examples.

  • 3.
    Blomberg, Niclas
    et al.
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Rojas, Cristian R.
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Wahlberg, Bo
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Regularization Paths for Re-Weighted Nuclear Norm Minimization2015In: IEEE Signal Processing Letters, ISSN 1070-9908, E-ISSN 1558-2361, Vol. 22, no 11, p. 1980-1984Article in journal (Refereed)
    Abstract [en]

    We consider a class of weighted nuclear norm optimization problems with important applications in signal processing, system identification, and model order reduction. The nuclear norm is commonly used as a convex heuristic for matrix rank constraints. Our objective is to minimize a quadratic cost subject to a nuclear norm constraint on a linear function of the decision variables, where the trade-off between the fit and the constraint is governed by a regularization parameter. The main contribution is an algorithm to determine the so-called approximate regularization path, which is the optimal solution up to a given error tolerance as a function of the regularization parameter. The advantage is that we only have to solve the optimization problem for a fixed number of values of the regularization parameter, with guaranteed error tolerance. The algorithm is exemplified on a weighted Hankel matrix model order reduction problem.

  • 4.
    Blomberg, Niclas
    et al.
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Rojas, Cristian
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Wahlberg, Bo
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Approximate Regularization Paths for Nuclear Norm Minimization using Singular Value Bounds: with Implementation and Extended Appendix2015Conference paper (Refereed)
    Abstract [en]

    The widely used nuclear norm heuristic for rank minimizationproblems introduces a regularization parameter which isdifficult to tune. We have recently proposed a method to approximatethe regularization path, i.e., the optimal solution asa function of the parameter, which requires solving the problemonly for a sparse set of points. In this paper, we extendthe algorithm to provide error bounds for the singular valuesof the approximation. We exemplify the algorithms on largescale benchmark examples in model order reduction. Here,the order of a dynamical system is reduced by means of constrainedminimization of the nuclear norm of a Hankel matrix.

  • 5. Ha, H.
    et al.
    Welsh, J. S.
    Blomberg, Niclas
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Rojas, Cristian R.
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Wahlberg, Bo
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Reweighted nuclear norm regularization: A SPARSEVA approach2015In: IFAC-PapersOnLine, ISSN 2405-8963, Vol. 48, no 28, p. 1172-1177Article in journal (Refereed)
    Abstract [en]

    The aim of this paper is to develop a method to estimate high order FIR and ARX models using least squares with re-weighted nuclear norm regularization. Typically, the choice of the tuning parameter in the reweighting scheme is computationally expensive, hence we propose the use of the SPARSEVA (SPARSe Estimation based on a VAlidation criterion) framework to overcome this problem. Furthermore, we suggest the use of the prediction error criterion (PEC) to select the tuning parameter in the SPARSEVA algorithm. Numerical examples demonstrate the veracity of this method which has close ties with the traditional technique of cross validation, but using much less computations.

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