David Luengo

Linear Latent Force Models Using Gaussian Processes

Purely data-driven approaches for machine learning present difficulties when data are scarce relative to the complexity of the model or when the model is forced to extrapolate. On the other hand, purely mechanistic approaches need to identify and specify all the interactions in the problem at hand (which may not be feasible) and still leave the issue of how to parameterize the system. In this paper, we present a hybrid approach using Gaussian processes and differential equations to combine data-driven modeling with a physical model of the system. We show how different, physically inspired, kernel functions can be developed through sensible, simple, mechanistic assumptions about the underlying system. The versatility of our approach is illustrated with three case studies from motion capture, computational biology, and geostatistics.

Efficient monte carlo methods for multi-dimensional learning with classifier chains

Multi-dimensional classification (MDC) is the supervised learning problem where an instance is associated with multiple classes, rather than with a single class, as in traditional classification problems. Since these classes are often strongly correlated, modeling the dependencies between them allows MDC methods to improve their performance - at the expense of an increased computational cost. In this paper we focus on the classifier chains (CC) approach for modeling dependencies, one of the most popular and highest-performing methods for multi-label classification (MLC), a particular case of MDC which involves only binary classes (i.e., labels). The original CC algorithm makes a greedy approximation, and is fast but tends to propagate errors along the chain. Here we present novel Monte Carlo schemes, both for finding a good chain sequence and performing efficient inference. Our algorithms remain tractable for high-dimensional data sets and obtain the best predictive performance across several real data sets. HighlightsA Monte Carlo approach for efficient classifier chains.Applied to learning from multi-label and multi-dimensional data.A theoretical and empirical study of payoff functions in the search space.An empirical cross-fold comparison with PCC and other related methods.

An Adaptive Population Importance Sampler: Learning From Uncertainty

Monte Carlo (MC) methods are well-known computational techniques, widely used in different fields such as signal processing, communications and machine learning. An important class of MC methods is composed of importance sampling (IS) and its adaptive extensions, such as population Monte Carlo (PMC) and adaptive multiple IS (AMIS). In this paper, we introduce a novel adaptive and iterated importance sampler using a population of proposal densities. The proposed algorithm, named adaptive population importance sampling (APIS), provides a global estimation of the variables of interest iteratively, making use of all the samples previously generated. APIS combines a sophisticated scheme to build the IS estimators (based on the deterministic mixture approach) with a simple temporal adaptation (based on epochs). In this way, APIS is able to keep all the advantages of both AMIS and PMC, while minimizing their drawbacks. Furthermore, APIS is easily parallelizable. The cloud of proposals is adapted in such a way that local features of the target density can be better taken into account compared to single global adaptation procedures. The result is a fast, simple, robust, and high-performance algorithm applicable to a wide range of problems. Numerical results show the advantages of the proposed sampling scheme in four synthetic examples and a localization problem in a wireless sensor network.

Bayesian estimation of chaotic signals generated by piecewise-linear maps

Chaotic signals are potentially attractive in a wide range of signal processing applications. This paper deals with Bayesian estimation of chaotic sequences generated by piecewise-linear (PWL) maps and observed in white Gaussian noise. The existence of invariant distributions associated with these sequences makes the development of Bayesian estimators quite natural, Both maximum a posteriori (MAP) and minimum mean square error (MS) estimators are derived. Computer simulations confirm the expected performance of both approaches, and show how the inclusion of a priori information produces in most cases an increase in performance over the maximum likelihood (ML) case.

Independent Doubly Adaptive Rejection Metropolis Sampling Within Gibbs Sampling

Bayesian methods have become very popular in signal processing lately, even though performing exact Bayesian inference is often unfeasible due to the lack of analytical expressions for optimal Bayesian estimators. In order to overcome this problem, Monte Carlo (MC) techniques are frequently used. Several classes of MC schemes have been developed, including Markov Chain Monte Carlo (MCMC) methods, particle filters and population Monte Carlo approaches. In this paper, we concentrate on the Gibbs-type approach, where automatic and fast samplers are needed to draw from univariate (full-conditional) densities. The Adaptive Rejection Metropolis Sampling (ARMS) technique is widely used within Gibbs sampling, but suffers from an important drawback: an incomplete adaptation of the proposal in some cases. In this work, we propose an alternative adaptive MCMC algorithm (IA2RMS) that overcomes this limitation, speeding up the convergence of the chain to the target, allowing us to simplify the construction of the sequence of proposals, and thus reducing the computational cost of the entire algorithm. Note that, although IA2RMS has been developed as an extremely efficient MCMC-within-Gibbs sampler, it also provides an excellent performance as a stand-alone algorithm when sampling from univariate distributions. In this case, the convergence of the proposal to the target is proved and a bound on the complexity of the proposal is provided. Numerical results, both for univariate (stand-alone IA2RMS) and multivariate (IA2RMS-within-Gibbs) distributions, show that IA2RMS outperforms ARMS and other classical techniques, providing a correlation among samples close to zero.Markov Chain Monte Carlo (MCMC) methods, such as the Metropolis-Hastings (MH) algorithm, are widely used for Bayesian inference. One of the most important challenges for any MCMC method is speeding up the convergence of the Markov chain, which depends crucially on a suitable choice of the proposal density. Adaptive Rejection Metropolis Sampling (ARMS) is a well-known MH scheme that generates samples from one-dimensional target densities making use of adaptive piecewise linear proposals constructed using support points taken from rejected samples. In this work we pinpoint a crucial drawback of the adaptive procedure used in ARMS: support points might never be added inside regions where the proposal is below the target. When this happens in many regions it leads to a poor performance of ARMS, and the sequence of proposals never converges to the target. In order to overcome this limitation, we propose two alternative adaptive schemes that guarantee convergence to the target distribution. These two new schemes improve the adaptive strategy of ARMS, thus allowing us to simplify the construction of the sequence of proposals. Numerical results show that the new algorithms outperform the standard ARMS and other techniques.

Efficient Multiple Importance Sampling Estimators

Multiple importance sampling (MIS) methods use a set of proposal distributions from which samples are drawn. Each sample is then assigned an importance weight that can be obtained according to different strategies. This work is motivated by the trade-off between variance reduction and computational complexity of the different approaches (classical vs. deterministic mixture) available for the weight calculation. A new method that achieves an efficient compromise between both factors is introduced in this letter. It is based on forming a partition of the set of proposal distributions and computing the weights accordingly. Computer simulations show the excellent performance of the associated partial deterministic mixture MIS estimator.

Layered adaptive importance sampling

Monte Carlo methods represent the de facto standard for approximating complicated integrals involving multidimensional target distributions. In order to generate random realizations from the target distribution, Monte Carlo techniques use simpler proposal probability densities to draw candidate samples. The performance of any such method is strictly related to the specification of the proposal distribution, such that unfortunate choices easily wreak havoc on the resulting estimators. In this work, we introduce a layered (i.e., hierarchical) procedure to generate samples employed within a Monte Carlo scheme. This approach ensures that an appropriate equivalent proposal density is always obtained automatically (thus eliminating the risk of a catastrophic performance), although at the expense of a moderate increase in the complexity. Furthermore, we provide a general unified importance sampling (IS) framework, where multiple proposal densities are employed and several IS schemes are introduced by applying the so-called deterministic mixture approach. Finally, given these schemes, we also propose a novel class of adaptive importance samplers using a population of proposals, where the adaptation is driven by independent parallel or interacting Markov chain Monte Carlo (MCMC) chains. The resulting algorithms efficiently combine the benefits of both IS and MCMC methods.

Scalable multi-output label prediction

Multi-output inference tasks, such as multi-label classification, have become increasingly important in recent years. A popular method for multi-label classification is classifier chains, in which the predictions of individual classifiers are cascaded along a chain, thus taking into account inter-label dependencies and improving the overall performance. Several varieties of classifier chain methods have been introduced, and many of them perform very competitively across a wide range of benchmark datasets. However, scalability limitations become apparent on larger datasets when modelling a fully cascaded chain. In particular, the methods? strategies for discovering and modelling a good chain structure constitute a major computational bottleneck. In this paper, we present the classifier trellis (CT) method for scalable multi-label classification. We compare CT with several recently proposed classifier chain methods to show that it occupies an important niche: it is highly competitive on standard multi-label problems, yet it can also scale up to thousands or even tens of thousands of labels. HighlightsA study of multi-output classification as graphical models.An empirical comparison of existing strategies for modelling dependency among outputs.A novel scalable approach based on a hill climbing heuristic: the classifier trellis.An empirical cross-fold comparison with other methods.A connection to structured output prediction and a comparison in a segmentation task.

Fully adaptive Gaussian mixture Metropolis-Hastings algorithm

Markov Chain Monte Carlo methods are widely used in signal processing and communications for statistical inference and stochastic optimization. In this work, we introduce an efficient adaptive Metropolis-Hastings algorithm to draw samples from generic multimodal and multidimensional target distributions. The proposal density is a mixture of Gaussian densities with all parameters (weights, mean vectors and covariance matrices) updated using all the previously generated samples applying simple recursive rules. Numerical results for the one and two-dimensional cases are provided.

Orthogonal parallel MCMC methods for sampling and optimization

Abstract Monte Carlo (MC) methods are widely used for Bayesian inference and optimization in statistics, signal processing and machine learning. A well-known class of MC methods are Markov Chain Monte Carlo (MCMC) algorithms. In order to foster better exploration of the state space, specially in high-dimensional applications, several schemes employing multiple parallel MCMC chains have been recently introduced. In this work, we describe a novel parallel interacting MCMC scheme, called orthogonal MCMC (O-MCMC), where a set of “vertical” parallel MCMC chains share information using some “horizontal” MCMC techniques working on the entire population of current states. More specifically, the vertical chains are led by random-walk proposals, whereas the horizontal MCMC techniques employ independent proposals, thus allowing an efficient combination of global exploration and local approximation. The interaction is contained in these horizontal iterations. Within the analysis of different implementations of O-MCMC, novel schemes in order to reduce the overall computational cost of parallel Multiple Try Metropolis (MTM) chains are also presented. Furthermore, a modified version of O-MCMC for optimization is provided by considering parallel Simulated Annealing (SA) algorithms. Numerical results show the advantages of the proposed sampling scheme in terms of efficiency in the estimation, as well as robustness in terms of independence with respect to initial values and the choice of the parameters.