arXiv subjects

Gianluigi Pillonetto

Publications and source records attributed to Gianluigi Pillonetto.

49 records · Page 3Linked to original sources

arXiv2014

Bayesian and regularization approaches to multivariable linear system identification: the role of rank penalties

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.

eess.SY↗

arXiv2014

Multi-agents adaptive estimation and coverage control using Gaussian regression

We consider a scenario where the aim of a group of agents is to perform the optimal coverage of a region according to a sensory function. In particular, centroidal Voronoi partitions have to be computed. The difficulty of the task is that the sensory function is unknown and has to be reconstructed on line from noisy measurements. Hence, estimation and coverage needs to be performed at the same time. We cast the problem in a Bayesian regression framework, where the sensory function is seen as a Gaussian random field. Then, we design a set of control inputs which try to well balance coverage and estimation, also discussing convergence properties of the algorithm. Numerical experiments show the effectivness of the new approach.

cs.MA↗

arXiv2013

Outlier robust system identification: a Bayesian kernel-based approach

In this paper, we propose an outlier-robust regularized kernel-based method for linear system identification. The unknown impulse response is modeled as a zero-mean Gaussian process whose covariance (kernel) is given by the recently proposed stable spline kernel, which encodes information on regularity and exponential stability. To build robustness to outliers, we model the measurement noise as realizations of independent Laplacian random variables. The identification problem is cast in a Bayesian framework, and solved by a new Markov Chain Monte Carlo (MCMC) scheme. In particular, exploiting the representation of the Laplacian random variables as scale mixtures of Gaussians, we design a Gibbs sampler which quickly converges to the target distribution. Numerical simulations show a substantial improvement in the accuracy of the estimates over state-of-the-art kernel-based methods.

stat.ML↗

arXiv2013

Kalman smoothing and block tridiagonal systems: new connections and numerical stability results

The Rauch-Tung-Striebel (RTS) and the Mayne-Fraser (MF) algorithms are two of the most popular smoothing schemes to reconstruct the state of a dynamic linear system from measurements collected on a fixed interval. Another (less popular) approach is the Mayne (M) algorithm introduced in his original paper under the name of Algorithm A. In this paper, we analyze these three smoothers from an optimization and algebraic perspective, revealing new insights on their numerical stability properties. In doing this, we re-interpret classic recursions as matrix decomposition methods for block tridiagonal matrices. First, we show that the classic RTS smoother is an implementation of the forward block tridiagonal (FBT) algorithm (also known as Thomas algorithm) for particular block tridiagonal systems. We study the numerical stability properties of this scheme, connecting the condition number of the full system to properties of the individual blocks encountered during standard recursion. Second, we study the M smoother, and prove it is equivalent to a backward block tridiagonal (BBT) algorithm with a stronger stability guarantee than RTS. Third, we illustrate how the MF smoother solves a block tridiagonal system, and prove that it has the same numerical stability properties of RTS (but not those of M). Finally, we present a new hybrid RTS/M (FBT/BBT) smoothing scheme, which is faster than MF, and has the same numerical stability guarantees of RTS and MF.

math.NA↗

arXiv2013

The connection between Bayesian estimation of a Gaussian random field and RKHS

Reconstruction of a function from noisy data is often formulated as a regularized optimization problem over an infinite-dimensional reproducing kernel Hilbert space (RKHS). The solution describes the observed data and has a small RKHS norm. When the data fit is measured using a quadratic loss, this estimator has a known statistical interpretation. Given the noisy measurements, the RKHS estimate represents the posterior mean (minimum variance estimate) of a Gaussian random field with covariance proportional to the kernel associated with the RKHS. In this paper, we provide a statistical interpretation when more general losses are used, such as absolute value, Vapnik or Huber. Specifically, for any finite set of sampling locations (including where the data were collected), the MAP estimate for the signal samples is given by the RKHS estimate evaluated at these locations.

stat.ML↗

arXiv2013

Sparse/Robust Estimation and Kalman Smoothing with Nonsmooth Log-Concave Densities: Modeling, Computation, and Theory

We introduce a class of quadratic support (QS) functions, many of which play a crucial role in a variety of applications, including machine learning, robust statistical inference, sparsity promotion, and Kalman smoothing. Well known examples include the l2, Huber, l1 and Vapnik losses. We build on a dual representation for QS functions using convex analysis, revealing the structure necessary for a QS function to be interpreted as the negative log of a probability density, and providing the foundation for statistical interpretation and analysis of QS loss functions. For a subclass of QS functions called piecewise linear quadratic (PLQ) penalties, we also develop efficient numerical estimation schemes. These components form a flexible statistical modeling framework for a variety of learning applications, together with a toolbox of efficient numerical methods for inference. In particular, for PLQ densities, interior point (IP) methods can be used. IP methods solve nonsmooth optimization problems by working directly with smooth systems of equations characterizing their optimality. The efficiency of the IP approach depends on the structure of particular applications. We consider the class of dynamic inverse problems using Kalman smoothing, where the aim is to reconstruct the state of a dynamical system with known process and measurement models starting from noisy output samples. In the classical case, Gaussian errors are assumed in the process and measurement models. The extended framework allows arbitrary PLQ densities to be used, and the proposed IP approach solves the generalized Kalman smoothing problem while maintaining the linear complexity in the size of the time series, just as in the Gaussian case. This extends the computational efficiency of classic algorithms to a much broader nonsmooth setting, and includes many recently proposed robust and sparse smoothers as special cases.

stat.ML↗

arXiv2013

Robust and Trend Following Student's t Kalman Smoothers

We present a Kalman smoothing framework based on modeling errors using the heavy tailed Student's t distribution, along with algorithms, convergence theory, open-source general implementation, and several important applications. The computational effort per iteration grows linearly with the length of the time series, and all smoothers allow nonlinear process and measurement models. Robust smoothers form an important subclass of smoothers within this framework. These smoothers work in situations where measurements are highly contaminated by noise or include data unexplained by the forward model. Highly robust smoothers are developed by modeling measurement errors using the Student's t distribution, and outperform the recently proposed L1-Laplace smoother in extreme situations with data containing 20% or more outliers. A second special application we consider in detail allows tracking sudden changes in the state. It is developed by modeling process noise using the Student's t distribution, and the resulting smoother can track sudden changes in the state. These features can be used separately or in tandem, and we present a general smoother algorithm and open source implementation, together with convergence analysis that covers a wide range of smoothers. A key ingredient of our approach is a technique to deal with the non-convexity of the Student's t loss function. Numerical results for linear and nonlinear models illustrate the performance of the new smoothers for robust and tracking applications, as well as for mixed problems that have both types of features.

math.OC↗

arXiv2013

Linear system identification using stable spline kernels and PLQ penalties

The classical approach to linear system identification is given by parametric Prediction Error Methods (PEM). In this context, model complexity is often unknown so that a model order selection step is needed to suitably trade-off bias and variance. Recently, a different approach to linear system identification has been introduced, where model order determination is avoided by using a regularized least squares framework. In particular, the penalty term on the impulse response is defined by so called stable spline kernels. They embed information on regularity and BIBO stability, and depend on a small number of parameters which can be estimated from data. In this paper, we provide new nonsmooth formulations of the stable spline estimator. In particular, we consider linear system identification problems in a very broad context, where regularization functionals and data misfits can come from a rich set of piecewise linear quadratic functions. Moreover, our anal- ysis includes polyhedral inequality constraints on the unknown impulse response. For any formulation in this class, we show that interior point methods can be used to solve the system identification problem, with complexity O(n3)+O(mn2) in each iteration, where n and m are the number of impulse response coefficients and measurements, respectively. The usefulness of the framework is illustrated via a numerical experiment where output measurements are contaminated by outliers.

stat.ML↗

arXiv2013

Optimization viewpoint on Kalman smoothing, with applications to robust and sparse estimation

In this paper, we present the optimization formulation of the Kalman filtering and smoothing problems, and use this perspective to develop a variety of extensions and applications. We first formulate classic Kalman smoothing as a least squares problem, highlight special structure, and show that the classic filtering and smoothing algorithms are equivalent to a particular algorithm for solving this problem. Once this equivalence is established, we present extensions of Kalman smoothing to systems with nonlinear process and measurement models, systems with linear and nonlinear inequality constraints, systems with outliers in the measurements or sudden changes in the state, and systems where the sparsity of the state sequence must be accounted for. All extensions preserve the computational efficiency of the classic algorithms, and most of the extensions are illustrated with numerical examples, which are part of an open source Kalman smoothing Matlab/Octave package.

math.OC↗

arXiv2013

Convex vs nonconvex approaches for sparse estimation: GLasso, Multiple Kernel Learning and Hyperparameter GLasso

The popular Lasso approach for sparse estimation can be derived via marginalization of a joint density associated with a particular stochastic model. A different marginalization of the same probabilistic model leads to a different non-convex estimator where hyperparameters are optimized. Extending these arguments to problems where groups of variables have to be estimated, we study a computational scheme for sparse estimation that differs from the Group Lasso. Although the underlying optimization problem defining this estimator is non-convex, an initialization strategy based on a univariate Bayesian forward selection scheme is presented. This also allows us to define an effective non-convex estimator where only one scalar variable is involved in the optimization process. Theoretical arguments, independent of the correctness of the priors entering the sparse model, are included to clarify the advantages of this non-convex technique in comparison with other convex estimators. Numerical experiments are also used to compare the performance of these approaches.

stat.ML↗

arXiv2011

Robust and Trend-following Kalman Smoothers using Student's t

We propose two nonlinear Kalman smoothers that rely on Student's t distributions. The T-Robust smoother finds the maximum a posteriori likelihood (MAP) solution for Gaussian process noise and Student's t observation noise, and is extremely robust against outliers, outperforming the recently proposed l1-Laplace smoother in extreme situations (e.g. 50% or more outliers). The second estimator, which we call the T-Trend smoother, is able to follow sudden changes in the process model, and is derived as a MAP solver for a model with Student's t-process noise and Gaussian observation noise. We design specialized methods to solve both problems which exploit the special structure of the Student's t-distribution, and provide a convergence theory. Both smoothers can be implemented with only minor modifications to an existing L2 smoother implementation. Numerical results for linear and nonlinear models illustrating both robust and fast tracking applications are presented.

stat.CO↗

arXiv2011

A statistical and computational theory for robust and sparse Kalman smoothing

Kalman smoothers reconstruct the state of a dynamical system starting from noisy output samples. While the classical estimator relies on quadratic penalization of process deviations and measurement errors, extensions that exploit Piecewise Linear Quadratic (PLQ) penalties have been recently proposed in the literature. These new formulations include smoothers robust with respect to outliers in the data, and smoothers that keep better track of fast system dynamics, e.g. jumps in the state values. In addition to L2, well known examples of PLQ penalties include the L1, Huber and Vapnik losses. In this paper, we use a dual representation for PLQ penalties to build a statistical modeling framework and a computational theory for Kalman smoothing. We develop a statistical framework by establishing conditions required to interpret PLQ penalties as negative logs of true probability densities. Then, we present a computational framework, based on interior-point methods, that solves the Kalman smoothing problem with PLQ penalties and maintains the linear complexity in the size of the time series, just as in the L2 case. The framework presented extends the computational efficiency of the Mayne-Fraser and Rauch-Tung-Striebel algorithms to a much broader non-smooth setting, and includes many known robust and sparse smoothers as special cases.

math.OC↗

arXiv2010

Client-server multi-task learning from distributed datasets

A client-server architecture to simultaneously solve multiple learning tasks from distributed datasets is described. In such architecture, each client is associated with an individual learning task and the associated dataset of examples. The goal of the architecture is to perform information fusion from multiple datasets while preserving privacy of individual data. The role of the server is to collect data in real-time from the clients and codify the information in a common database. The information coded in this database can be used by all the clients to solve their individual learning task, so that each client can exploit the informative content of all the datasets without actually having access to private data of others. The proposed algorithmic framework, based on regularization theory and kernel methods, uses a suitable class of mixed effect kernels. The new method is illustrated through a simulated music recommendation system.

cs.LG↗