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Research Grants:
Research Grants:
Israel Science Foundation: Personal Grant: 2406/22
Young Faculty Chair: Jane and Larry Sherman Fellowship
Variation diminishing systems
Collaborators: Rodolphe Sepulchre, Tobias Damm, Thiago B. Burghi, Somayeh Sojoudi, Chaim Roth, Kang TongVariation diminishing systems
The so-called variation diminishing property is a fundamental component in many areas such as statistic, approximation and interpolation theory, filtering, etc. In this project, we provide a system theoretical approach to these high-level linear algebra tools as means to study monotone dynamical systems. In particular, we attempt to provide bridges between scaleable positive systems theory, model order reduction, optimization algorithms and learning theory.
Publications:
C. Roth, C. Grussler, On System Operators with Variation Bounding Properties, arXiv, submitted for review, 2024
C. Roth, C. Grussler, On Variation Bounding System Operators, 22nd European Control Conference (ECC), pp. 2138-2142, Stockholm, Sweden, 2024
C. Grussler, T. B. Burghi, On the Monotonicity of Frequency Response Gains, 62nd IEEE Conference on Decision and Control (CDC), pp. 1685-1691, Singapore, Singapore, 2023
C. Grussler, T. B. Burghi, Somayeh Sojoudi, Internally Hankel k-positive systems, SIAM J. Control Optim., 60(4), 2373-2392, 2022
C. Grussler, R. Sepulchre, Variation diminishing linear time-invariant systems, vol. 136, pp. 109985, Automatica, 2022
C. Grussler, T. Damm, R. Sepulchre, Balanced truncation of k-positive systems, IEEE Trans. Autom. Control, 67(1), pp. 526-531, 2022
C. Grussler, R. Sepulchre, Variation diminishing Hankel operators, 59th IEEE Conference on Decision and Control (CDC), pp. 4529-4534, Jeju, Korea (South), 2020
C. Grussler, R. Sepulchre, Strongly unimodal systems, 18th European Control Conference (ECC), pp. 3273-3278, Naples, Italy, 2019
Low-rank optimization with convex constraints
Collaborators: Anders Rantzer, Pontus Giselsson, Armin Zare, Mihailo Jovanović, Maya MarmaryLow-rank optimization with convex constraints
The desire to find a low-rank/sparse solution that fulfils certain constraints is common in many areas that are driven by data. For example, the completion of unknown entries in a matrix with low-rank constraint was demonstrated to work successfully in the famous Netflix challenge. Or low-rank approximation of a Hankel-matrix under the preservation of the Hankel-structure can be used to identify reduced order models. Typically, heuristics like the nuclear-norm regularisation method are the state of the art to address such problems, which leads to suboptimal solutions.
Our goal is to fill this gap and give deterministic optimal solutions that do not depend on a regularisation parameter. To this end, we developed the tool of so-called "low-rank inducing norms". Those norms are SDP-representable, have cheap computational proximal mappings and provide an a posterior optimality certificate. The later can be of great interest when evaluating the performance of algorithms that lack converge guarantees to a global optimum. In particular, these norms allow us to gain insights in the convergence of so-called non-convex proximal splitting methods.
As a result, we can use our findings for problem such as:
PCA with convex constraints
Low-rank linear regression
System identification and model order reduction
Low-rank matrix Completion
Software for low-rank inducing norms is available for:
Software for low-rank inducing norms is available for:
Publications:
Publications:
C. Grussler, P. Giselsson, Efficient proximal mapping computation for unitarily invariant low-rank inducing norms, J. Optim. Theory. Appl., pp. 1573-2878, 2021
C. Grussler, P. Giselsson, Optimality interpretations for a atomic norms, 18th European Control Conference (ECC), pp. 1473-1477, Naples, Italy, 2019
C. Grussler, P. Giselsson, Low-rank inducing norms with optimality interpretations, SIAM J. Optim., 28(4), 3057–3078, 2018
C. Grussler, A. Rantzer, P. Giselsson, Low-rank optimization with convex constraints, IEEE Trans. Autom. Control, 63(11), 4000–4007, 2018
C. Grussler, P. Giselsson, Local convergence of proximal splitting methods for rank constrained problems, 56th IEEE Conference on Decision and Control (CDC), pp. 702-708, Melbourne, VIC, Australia, 2017
C. Grussler, A. Zare, M. R. Jovanović, A. Rantzer, The use of the r* heuristic in covariance completion problems, 55th IEEE Conference on Decision and Control (CDC), pp. 1978-1983, Las Vegas, NV, 2016
C. Grussler, A. Rantzer, On optimal low-rank approximation of non-negative matrices, 54th IEEE Conference on Decision and Control (CDC), pp. 5278-5283, Osaka, Japan, 2015
Model reduction & system identification of positive systems
Collaborators: Anders Rantzer, Tobias Damm, Jack Umenberger, Ian Manchester, Sara-Lea DahanModel reduction & system identification of positive systems
Model order reduction has been used for decades to approximately describe complex systems by simple models. However, the known techniques often violate the simple constraint that some physical quantities can never occur in a negative amount. The property of being positive from input to output is commonly called external positivity. Transportation networks, biological systems as well as the classical heat transfer model are only a few examples for such systems.
Therefore, we seek to modify existing and to develop new methods, which preserve this positivity constraint. Research in this direction has been conducted earlier, but with strong conservatism regarding dimensionality and errors. Our goal is to supply new approximation strategies with the incentive of weakening the current conservatism. To this end, we developed a modified version of balanced truncation on system that leave ellipsoidal cones invariant, where ellipsoidal cone invariance is used as a sufficient certificate for external positivity.
Finally, also in system identification we often seek to learn an externally positive model. Since system identification is closely related to the problem of model order reduction, we can use similar techniques to guarantee the identification of an externally positive system.
Publications:
Publications:
C. Grussler, A. Rantzer, On the Similarity of Nonnegative and Metzler Hessenberg Forms, Special Matrices, vol. 10, no. 1, pp. 1-8., 2022
C. Grussler, A. Rantzer, On second-order cone positive systems, SIAM J. Control Optim., 59(4), 2717–2739, 2021
C. Grussler, J. Umenberger, I. Manchester, Identification of externally positive systems, 56th IEEE Conference on Decision and Control (CDC), pp. 6549-6554, Melbourne, VIC, Australia, 2017
C. Grussler, A. Rantzer, Modified balanced truncation preserving ellipsoidal cone-invariance, 53rd IEEE Conference on Decision and Control (CDC), pp. 2365-2370, Los Angeles, CA, 2014
C. Grussler, T. Damm, A symmetry approach for balanced truncation of positive linear systems, 51st IEEE Conference on Decision and Control (CDC), pp. 4308-4313, Maui, HI, 2012
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