Recent developments in linear system identification have proposed the use ofnon-parameteric methods, relying on regularization strategies, to handle theso-called bias/variance trade-off. This paper introduces an impulse responseestimator which relies on an $ell_2$-type regularization including arank-penalty derived using the log-det heuristic as a smooth approximation tothe rank function. This allows to account for different properties of theestimated impulse response (e.g. smoothness and stability) while alsopenalizing high-complexity models. This also allows to account and enforcecoupling between different input-output channels in MIMO systems. According tothe Bayesian paradigm, the parameters defining the relative weight of the tworegularization terms as well as the structure of the rank penalty are estimatedoptimizing the marginal likelihood. Once these hyperameters have beenestimated, the impulse response estimate is available in closed form.Experiments show that the proposed method is superior to the estimator relyingon the "classic" $ell_2$-regularization alone as well as those based in atomicand nuclear norm.
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