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