One central theme in machine learning is function estimation from sparse and noisy data. An example is supervised learning where the elements of the training set are couples, each containing an input location and an output response. In the last decades, a substantial amount of work has been devoted to design estimators for the unknown function and to study their convergence to the optimal predictor, also characterizing the learning rate. These results typically rely on stationary assumptions where input locations are drawn from a probability distribution that does not change in time. In this work, we consider kernel-based ridge regression and derive convergence conditions under non stationary distributions, addressing also cases where stochastic adaption may happen infinitely often. This includes the important exploration-exploitation problems where e.g., a set of agents/robots has to monitor an environment to reconstruct a sensorial field and their movements rules are continuously updated on the basis of the acquired knowledge on the field and/or the surrounding environment.

Estimating a set of orthogonal functions from a finite set of noisy data plays a crucial role in several areas such as imaging, dictionary learning and compressed sensing. The problem turns out especially hard due to its intrinsic non-convexity. In this paper, we solve it by recasting it in the framework of multi-task learning in Hilbert spaces, where orthogonality plays a role as inductive bias. Two perspectives are analyzed. The first one is mainly theoretic. It considers a formulation of the problem where non-orthogonal function estimates are seen as noisy data belonging to an infinite-dimensional space from which orthogonal functions have to be reconstructed. We then provide results concerning the existence and the convergence of the optimizers. The second one is more oriented towards applications. It consists in a learning scheme where orthogonal functions are directly inferred from a finite amount of noisy data. It relies on regularization in reproducing kernel Hilbert spaces and on the introduction of special penalty terms promoting orthogonality among tasks. The problem is then cast in a Bayesian framework, overcoming non-convexity through an efficient Markov chain Monte Carlo scheme. If orthogonality is not certain, our scheme can also understand from data if such form of task interaction really holds.

Reconstruction of a function from noisy data is key in machine learning and is often formulated as a regularized optimization problem over an infinite-dimensional reproducing kernel Hilbert space (RKHS). The solution suitably balances adherence to the observed data and the corresponding 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 brief, 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 (that includes where the data were collected), the maximum a posteriori estimate for the signal samples is given by the RKHS estimate evaluated at the sampling locations. This connection establishes a firm statistical foundation for several stochastic approaches used to estimate unknown regularization parameters. To illustrate this, we develop a numerical scheme that implements a Bayesian estimator with an absolute value loss. This estimator is used to learn a function from measurements contaminated by outliers.

This paper considers the classification problem using support vector (SV) machines and investigates how to maximally reduce the size of the training set without losing information. Under separable data set assumptions, we derive the exact conditions stating which observations can be discarded without diminishing the overall information content. For this purpose, we introduce the concept of potential SVs, i.e., those data that can become SVs when future data become available. To complement this, we also characterize the set of discardable vectors (DVs), i.e., those data that, given the current data set, can never become SVs. Thus, these vectors are useless for future training purposes and can eventually be removed without loss of information. Then, we provide an efficient algorithm based on linear programming that returns the potential and DVs by constructing a simplex tableau. Finally, we compare it with alternative algorithms available in the literature on some synthetic data as well as on data sets from standard repositories.

We address the problem of compressing large and distributed signals monitored by a Wireless Sensor Network (WSN) and recovering them through the collection of a small number of samples. We propose a sparsity model that allows the use of Compressive Sensing (CS) for the online recovery of large data sets in real WSN scenarios, exploiting Principal Component Analysis (PCA) to capture the spatial and temporal characteristics of real signals. Bayesian analysis is utilized to approximate the statistical distribution of the principal components and to show that the Laplacian distribution provides an accurate representation of the statistics of real data. This combined CS and PCA technique is subsequently integrated into a novel framework, namely, SCoRe1: Sensing, Compression and Recovery through ON-line Estimation for WSNs. SCoRe1 is able to effectively self-adapt to unpredictable changes in the signal statistics thanks to a feedback control loop that estimates, in real time, the signal reconstruction error. We also propose an extensive validation of the framework used in conjunction with CS as well as with standard interpolation techniques, testing its performance for real world signals. The results in this paper have the merit of shedding new light on the performance limits of CS when used as a recovery tool in WSNs.

A client-server architecture to simultaneously solve multiple learning tasks from distributed datasets is described. In such architecture, each client corresponds to 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. Such information can be used by all the clients to solve their individual learning task, so that each client can exploit the information content of all the datasets without actually having access to private data of others. The proposed algorithmic framework, based on regularization and kernel methods, uses a suitable class of "mixed effect" kernels. The methodology is illustrated through a simulated recommendation system, as well as an experiment involving pharmacological data coming from a multicentric clinical trial.

We present a moving horizon approach for estimating the state of a nonlinear dynamic system that may be subject to inequality constraints. The method takes advantage of a recent smoothing algorithm proposed in the literature based on interior point techniques. The approach exploits the same decomposition used for unconstrained Kalman-Bucy smoothers. Hence, the number of operations required by the algorithm scales linearly with the length of the horizon, making it suitable for online applications. We apply this method to the robot localization problem, showing that it is able to produce much more accurate results than the iterated Kalman filter with little additional computational effort.

We introduce a new Bayesian nonparametric approach to identification of sparse dynamic linear systems. The impulse responses are modeled as Gaussian processes whose autocovariances encode the BIBO stability constraint, as defined by the recently introduced “Stable Spline kernel”. Sparse solutions are obtained by placing exponential hyperpriors on the scale factors of such kernels. Numerical experiments regarding estimation of ARMAX models show that this technique provides a definite advantage over a group LAR algorithm and state-of-the-art parametric identification techniques based on prediction error minimization.

Standard single-task kernel methods have recently been extended to the case of multitask learning in the context of regularization theory. There are experimental results, especially in biomedicine, showing the benefit of the multitask approach compared to the single-task one. However, a possible drawback is computational complexity. For instance, when regularization networks are used, complexity scales as the cube of the overall number of training data, which may be large when several tasks are involved. The aim of this paper is to derive an efficient computational scheme for an important class of multitask kernels. More precisely, a quadratic loss is assumed and each task consists of the sum of a common term and a task-specific one. Within a Bayesian setting, a recursive online algorithm is obtained, which updates both estimates and confidence intervals as new data become available. The algorithm is tested on two simulated problems and a real data set relative to xenobiotics administration in human patients.

This paper presents an experimental validation of an online estimation algorithm we recently investigated theoretically. One of the peculiar characteristics of the approach we propose is the ability to perform an online estimation of the variance parameters that regulate the dynamics of the nonlinear dynamical model used. The approach exploits and extends classical iterated Kalman filtering equations by propagating an approximation of the marginal posterior of the unknown variances over time. The method has been previously used to model and solve a localization task for multiple robots equipped only with a sensor returning mutual distances. In this paper we present a first experimental validation of the algorithm that complements and confirms our initial promising theoretical findings. Our current implementation relies on a sensor returning distance estimates based on a simple image processing algorithm. Such sensor is inherently and intentionally noisy, and in this study we show that our technique is capable of appropriately estimating the variance describing the noise affecting this sensor. We conclude proving experimentally that the procedure we present ensures a performance comparable to similar algorithms that require significantly more a priori information.

The multirobot localization problem is solved in this paper using an innovative approach related to Tikhonov regularization. We release the requirement that robots are equipped with sensors to estimate their own motion, as well as the requirement that covariance matrices describing system and measure noises are perfectly known. Robots are assumed to have a single sensor returning noisy measurements of mutual distances while they move along unknown paths. The proposed algorithm estimates online both the robots’ poses as well as the unknown covariance parameters. In addition to the classical iterations of the well known iterated Kalman filter, we include iterations that propagate an approximation of the posterior marginal densities of the unknown variances. Simulationl results provide evidence that the algorithm is capable of accurately estimating the variances online while at the same time keeping the localization error bounded.

We propose a novel single-shot motion-planning algorithm based on adaptive random walks. The proposed algorithm turns out to be simple to implement, and the solution it produces can be easily and efficiently optimized. Furthermore, the algorithm can incorporate adaptive components, so the developer is not required to specify all the parameters of the random distributions involved, and the algorithm itself can adapt to the environment it is moving in. Proofs of the theoretical soundness of the algorithm are provided, as well as implementation details. Numerical comparisons with well-known algorithms illustrate its effectiveness.

We propose a novel motion planning algorithm based on adaptive random walks. The proposed algorithm turns out to be easy to implement and the solution it produces can be easily and efficiently optimized. Furthermore the algorithm can incorporate adaptive components, so that the developer is not required to specify all the parameters of the random distributions involved, and the algorithm itself can adapt to the environment it is moving in. Proofs of the theoretical soundness of the algorithm are provided as well as implementation details. Numerical comparisons with well known algorithms illustrate its effectiveness.