Regularization and Bayesian Learning in Dynamical Systems: Past, Present and Future

Regularization and Bayesian methods for system identification have been repopularized in the recent years, and proved to be competitive w.r.t. classical parametric approaches. In this paper we shall make an attempt to illustrate how the use of regularization in system identification has evolved over the years, starting from the early contributions both in the Automatic Control as well as Econometrics and Statistics literature. In particular we shall discuss some fundamental issues such as compound estimation problems and exchangeability which play and important role in regularization and Bayesian approaches, as also illustrated in early publications in Statistics. The historical and foundational issues will be given more emphasis (and space), at the expense of the more recent developments which are only briefly discussed. The main reason for such a choice is that, while the recent literature is readily available, and surveys have already been published on the subject, in the author's opinion a clear link with past work had not been completely clarified.Comment: Plenary Presentation at the IFAC SYSID 2015. Submitted to Annual Reviews in Contro

High-Dimensional Regression with Gaussian Mixtures and Partially-Latent Response Variables

In this work we address the problem of approximating high-dimensional data with a low-dimensional representation. We make the following contributions. We propose an inverse regression method which exchanges the roles of input and response, such that the low-dimensional variable becomes the regressor, and which is tractable. We introduce a mixture of locally-linear probabilistic mapping model that starts with estimating the parameters of inverse regression, and follows with inferring closed-form solutions for the forward parameters of the high-dimensional regression problem of interest. Moreover, we introduce a partially-latent paradigm, such that the vector-valued response variable is composed of both observed and latent entries, thus being able to deal with data contaminated by experimental artifacts that cannot be explained with noise models. The proposed probabilistic formulation could be viewed as a latent-variable augmentation of regression. We devise expectation-maximization (EM) procedures based on a data augmentation strategy which facilitates the maximum-likelihood search over the model parameters. We propose two augmentation schemes and we describe in detail the associated EM inference procedures that may well be viewed as generalizations of a number of EM regression, dimension reduction, and factor analysis algorithms. The proposed framework is validated with both synthetic and real data. We provide experimental evidence that our method outperforms several existing regression techniques

Analyzing sparse dictionaries for online learning with kernels

Many signal processing and machine learning methods share essentially the same linear-in-the-parameter model, with as many parameters as available samples as in kernel-based machines. Sparse approximation is essential in many disciplines, with new challenges emerging in online learning with kernels. To this end, several sparsity measures have been proposed in the literature to quantify sparse dictionaries and constructing relevant ones, the most prolific ones being the distance, the approximation, the coherence and the Babel measures. In this paper, we analyze sparse dictionaries based on these measures. By conducting an eigenvalue analysis, we show that these sparsity measures share many properties, including the linear independence condition and inducing a well-posed optimization problem. Furthermore, we prove that there exists a quasi-isometry between the parameter (i.e., dual) space and the dictionary's induced feature space.Comment: 10 page