Maximum Entropy Vector Kernels for MIMO system identification

Recent contributions have framed linear system identification as a nonparametric regularized inverse problem. Relying on $\ell_2$-type regularization which accounts for the stability and smoothness of the impulse response to be estimated, these approaches have been shown to be competitive w.r.t classical parametric methods. In this paper, adopting Maximum Entropy arguments, we derive a new $\ell_2$ penalty deriving from a vector-valued kernel; to do so we exploit the structure of the Hankel matrix, thus controlling at the same time complexity, measured by the McMillan degree, stability and smoothness of the identified models. As a special case we recover the nuclear norm penalty on the squared block Hankel matrix. In contrast with previous literature on reweighted nuclear norm penalties, our kernel is described by a small number of hyper-parameters, which are iteratively updated through marginal likelihood maximization; constraining the structure of the kernel acts as a (hyper)regularizer which helps controlling the effective degrees of freedom of our estimator. To optimize the marginal likelihood we adapt a Scaled Gradient Projection (SGP) algorithm which is proved to be significantly computationally cheaper than other first and second order off-the-shelf optimization methods. The paper also contains an extensive comparison with many state-of-the-art methods on several Monte-Carlo studies, which confirms the effectiveness of our procedure

Hybrid Beamforming via the Kronecker Decomposition for the Millimeter-Wave Massive MIMO Systems

Despite its promising performance gain, the realization of mmWave massive MIMO still faces several practical challenges. In particular, implementing massive MIMO in the digital domain requires hundreds of RF chains matching the number of antennas. Furthermore, designing these components to operate at the mmWave frequencies is challenging and costly. These motivated the recent development of hybrid-beamforming where MIMO processing is divided for separate implementation in the analog and digital domains, called the analog and digital beamforming, respectively. Analog beamforming using a phase array introduces uni-modulus constraints on the beamforming coefficients, rendering the conventional MIMO techniques unsuitable and call for new designs. In this paper, we present a systematic design framework for hybrid beamforming for multi-cell multiuser massive MIMO systems over mmWave channels characterized by sparse propagation paths. The framework relies on the decomposition of analog beamforming vectors and path observation vectors into Kronecker products of factors being uni-modulus vectors. Exploiting properties of Kronecker mixed products, different factors of the analog beamformer are designed for either nulling interference paths or coherently combining data paths. Furthermore, a channel estimation scheme is designed for enabling the proposed hybrid beamforming. The scheme estimates the AoA of data and interference paths by analog beam scanning and data-path gains by analog beam steering. The performance of the channel estimation scheme is analyzed. In particular, the AoA spectrum resulting from beam scanning, which displays the magnitude distribution of paths over the AoA range, is derived in closed-form. It is shown that the inter-cell interference level diminishes inversely with the array size, the square root of pilot sequence length and the spatial separation between paths.Comment: Submitted to IEEE JSAC Special Issue on Millimeter Wave Communications for Future Mobile Networks, minor revisio

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

Regularized linear system identification using atomic, nuclear and kernel-based norms: the role of the stability constraint

Inspired by ideas taken from the machine learning literature, new regularization techniques have been recently introduced in linear system identification. In particular, all the adopted estimators solve a regularized least squares problem, differing in the nature of the penalty term assigned to the impulse response. Popular choices include atomic and nuclear norms (applied to Hankel matrices) as well as norms induced by the so called stable spline kernels. In this paper, a comparative study of estimators based on these different types of regularizers is reported. Our findings reveal that stable spline kernels outperform approaches based on atomic and nuclear norms since they suitably embed information on impulse response stability and smoothness. This point is illustrated using the Bayesian interpretation of regularization. We also design a new class of regularizers defined by "integral" versions of stable spline/TC kernels. Under quite realistic experimental conditions, the new estimators outperform classical prediction error methods also when the latter are equipped with an oracle for model order selection

The edge cloud: A holistic view of communication, computation and caching

The evolution of communication networks shows a clear shift of focus from just improving the communications aspects to enabling new important services, from Industry 4.0 to automated driving, virtual/augmented reality, Internet of Things (IoT), and so on. This trend is evident in the roadmap planned for the deployment of the fifth generation (5G) communication networks. This ambitious goal requires a paradigm shift towards a vision that looks at communication, computation and caching (3C) resources as three components of a single holistic system. The further step is to bring these 3C resources closer to the mobile user, at the edge of the network, to enable very low latency and high reliability services. The scope of this chapter is to show that signal processing techniques can play a key role in this new vision. In particular, we motivate the joint optimization of 3C resources. Then we show how graph-based representations can play a key role in building effective learning methods and devising innovative resource allocation techniques.Comment: to appear in the book "Cooperative and Graph Signal Pocessing: Principles and Applications", P. Djuric and C. Richard Eds., Academic Press, Elsevier, 201