This paper proposes a novel scheme for reduced-rank Gaussian process regression. The method is based on an approximate series expansion of the covariance function in terms of an eigenfunction expansion of the Laplace operator in a compact subset of $\mathbb{R}^d$. On this approximate eigenbasis the eigenvalues of the covariance function can be expressed as simple functions of the spectral density of the Gaussian process, which allows the GP inference to be solved under a computational cost scaling as $\mathcal{O}(nm^2)$ (initial) and $\mathcal{O}(m^3)$ (hyperparameter learning) with $m$ basis functions and $n$ data points. The approach also allows for rigorous error analysis with Hilbert space theory, and we show that the approximation becomes exact when the size of the compact subset and the number of eigenfunctions go to infinity. The expansion generalizes to Hilbert spaces with an inner product which is defined as an integral over a specified input density. The method is compared to previously proposed methods theoretically and through empirical tests with simulated and real data.

The recent developments on the syntactical and morphological analysis of natural language text constitute the first step towards a more ambitious goal, that of assigning a proper form of meaning to arbitrary text compounds. Indeed, for certain really "intelligent" applications, such as machine translation, question-answering systems, paraphrase detection, or automatic essay scoring, to name just a few, there will always exist a gap between raw linguistic information (such as part-of-speech labels, for example) and the knowledge of the real world that is needed for the completion of the task in a satisfactory way. Semantic analysis has exactly this role, aiming to close (or reduce as much as possible) this gap by linking the linguistic information with semantic representations that embody this elusive real-world knowledge. The traditional way of adding semantics to sentences is a syntax-driven compositional approach: every word in the sentence is associated with a primitive symbol or a predicate, and these are combined to larger and larger logical forms based on the syntactical rules of the grammar. At the end of the syntactical analysis, the logical representation of the whole sentence is a complex formula that can be fed to a theorem prover for further processing. Although such an approach seems intuitive, it has been shown that it is rather inefficient for any practical application (for example, Bos and Markert (2006) get very low recall scores for a textual entailment task).

We present two graph-based algorithms for multiclass segmentation of high-dimensional data. The algorithms use a diffuse interface model based on the Ginzburg-Landau functional, related to total variation compressed sensing and image processing. A multiclass extension is introduced using the Gibbs simplex, with the functional's double-well potential modified to handle the multiclass case. The first algorithm minimizes the functional using a convex splitting numerical scheme. The second algorithm is a uses a graph adaptation of the classical numerical Merriman-Bence-Osher (MBO) scheme, which alternates between diffusion and thresholding. We demonstrate the performance of both algorithms experimentally on synthetic data, grayscale and color images, and several benchmark data sets such as MNIST, COIL and WebKB. We also make use of fast numerical solvers for finding the eigenvectors and eigenvalues of the graph Laplacian, and take advantage of the sparsity of the matrix. Experiments indicate that the results are competitive with or better than the current state-of-the-art multiclass segmentation algorithms.

Modern data acquisition routinely produces massive amounts of high dimensional data with complex statistical dependency structures. Latent variable graphical models provide a succinct representation of such complex dependency structures by relating the observed variables to a set of latent ones. By defining a joint distribution over observed and latent variables, the marginal distribution of the observed variables can be obtained by integrating out the latent ones. This allows complex distributions over observed variables (e.g., clique models) to be expressed in terms of more tractable joint models (e.g., tree models) over the augmented variable space. Probabilistic graphical models with latent variables have been deployed successfully to a diverse range of problems such as in document analysis (Blei et al., 2002), social network modeling (Hoff et al., 2002), speech recognition (Rabiner and Juang, 1986) and bioinformatics (Clark, 1990). In this paper, we focus on latent variable models where the latent structures are trees (we call it a "latent tree" for short).

Local search algorithms applied to optimization problems often suffer from getting trapped in a local optimum. The common solution for this deficiency is to restart the algorithm when no progress is observed. Alternatively, one can start multiple instances of a local search algorithm, and allocate computational resources (in particular, processing time) to the instances depending on their behavior. Hence, a multi-start strategy has to decide (dynamically) when to allocate additional resources to a particular instance and when to start new instances. In this paper we propose multi-start strategies motivated by works on multi-armed bandit problems and Lipschitz optimization with an unknown constant. The strategies continuously estimate the potential performance of each algorithm instance by supposing a convergence rate of the local search algorithm up to an unknown constant, and in every phase allocate resources to those instances that could converge to the optimum for a particular range of the constant. Asymptotic bounds are given on the performance of the strategies. In particular, we prove that at most a quadratic increase in the number of times the target function is evaluated is needed to achieve the performance of a local search algorithm started from the attraction region of the optimum. Experiments are provided using SPSA (Simultaneous Perturbation Stochastic Approximation) and k-means as local search algorithms, and the results indicate that the proposed strategies work well in practice, and, in all cases studied, need only logarithmically more evaluations of the target function as opposed to the theoretically suggested quadratic increase.

Time series are ubiquitous, and a measure to assess their similarity is a core part of many computational systems. In particular, the similarity measure is the most essential ingredient of time series clustering and classification systems. Because of this importance, countless approaches to estimate time series similarity have been proposed. However, there is a lack of comparative studies using empirical, rigorous, quantitative, and large-scale assessment strategies. In this article, we provide an extensive evaluation of similarity measures for time series classification following the aforementioned principles. We consider 7 different measures coming from alternative measure `families', and 45 publicly-available time series data sets coming from a wide variety of scientific domains. We focus on out-of-sample classification accuracy, but in-sample accuracies and parameter choices are also discussed. Our work is based on rigorous evaluation methodologies and includes the use of powerful statistical significance tests to derive meaningful conclusions. The obtained results show the equivalence, in terms of accuracy, of a number of measures, but with one single candidate outperforming the rest. Such findings, together with the followed methodology, invite researchers on the field to adopt a more consistent evaluation criteria and a more informed decision regarding the baseline measures to which new developments should be compared.

The need for efficient monitoring of spatio-temporal dynamics in large environmental applications, such as the water quality monitoring in rivers and lakes, motivates the use of robotic sensors in order to achieve sufficient spatial coverage. Typically, these robots have bounded resources, such as limited battery or limited amounts of time to obtain measurements. Thus, careful coordination of their paths is required in order to maximize the amount of information collected, while respecting the resource constraints. In this paper, we present an efficient approach for near-optimally solving the NP-hard optimization problem of planning such informative paths. In particular, we first develop eSIP (efficient Single-robot Informative Path planning), an approximation algorithm for optimizing the path of a single robot. Hereby, we use a Gaussian Process to model the underlying phenomenon, and use the mutual information between the visited locations and remainder of the space to quantify the amount of information collected. We prove that the mutual information collected using paths obtained by using eSIP is close to the information obtained by an optimal solution. We then provide a general technique, sequential allocation, which can be used to extend any single robot planning algorithm, such as eSIP, for the multi-robot problem. This procedure approximately generalizes any guarantees for the single-robot problem to the multi-robot case. We extensively evaluate the effectiveness of our approach on several experiments performed in-field for two important environmental sensing applications, lake and river monitoring, and simulation experiments performed using several real world sensor network data sets.

The paper investigates parameterized approximate message-passing schemes that are based on bounded inference and are inspired by Pearl's belief propagation algorithm (BP). We start with the bounded inference mini-clustering algorithm and then move to the iterative scheme called Iterative Join-Graph Propagation (IJGP), that combines both iteration and bounded inference. Algorithm IJGP belongs to the class of Generalized Belief Propagation algorithms, a framework that allowed connections with approximate algorithms from statistical physics and is shown empirically to surpass the performance of mini-clustering and belief propagation, as well as a number of other state-of-the-art algorithms on several classes of networks. We also provide insight into the accuracy of iterative BP and IJGP by relating these algorithms to well known classes of constraint propagation schemes.

The Weighted Constraint Satisfaction Problem (WCSP) framework allows representing and solving problems involving both hard constraints and cost functions. It has been applied to various problems, including resource allocation, bioinformatics, scheduling, etc. To solve such problems, solvers usually rely on branch-and-bound algorithms equipped with local consistency filtering, mostly soft arc consistency. However, these techniques are not well suited to solve problems with very large domains. Motivated by the resolution of an RNA gene localization problem inside large genomic sequences, and in the spirit of bounds consistency for large domains in crisp CSPs, we introduce soft bounds arc consistency, a new weighted local consistency specifically designed for WCSP with very large domains. Compared to soft arc consistency, BAC provides significantly improved time and space asymptotic complexity. In this paper, we show how the semantics of cost functions can be exploited to further improve the time complexity of BAC. We also compare both in theory and in practice the efficiency of BAC on a WCSP with bounds consistency enforced on a crisp CSP using cost variables. On two different real problems modeled as WCSP, including our RNA gene localization problem, we observe that maintaining bounds arc consistency outperforms arc consistency and also improves over bounds consistency enforced on a constraint model with cost variables.

We investigate the computational complexity of testing dominance and consistency in CP-nets. Previously, the complexity of dominance has been determined for restricted classes in which the dependency graph of the CP-net is acyclic. However, there are preferences of interest that define cyclic dependency graphs; these are modeled with general CP-nets. In our main results, we show here that both dominance and consistency for general CP-nets are PSPACE-complete. We then consider the concept of strong dominance, dominance equivalence and dominance incomparability, and several notions of optimality, and identify the complexity of the corresponding decision problems. The reductions used in the proofs are from STRIPS planning, and thus reinforce the earlier established connections between both areas.