Gaussian belief propagation (GaBP) is an iterative algorithm for computing the mean of a multivariate Gaussian distribution, or equivalently, the minimum of a multivariate positive definite quadratic function. Sufficient conditions, such as walk-summability, that guarantee the convergence and correctness of GaBP are known, but GaBP may fail to converge to the correct solution given an arbitrary positive definite quadratic function. As was observed in previous work, the GaBP algorithm fails to converge if the computation trees produced by the algorithm are not positive definite. In this work, we will show that the failure modes of the GaBP algorithm can be understood via graph covers, and we prove that a parameterized generalization of the min-sum algorithm can be used to ensure that the computation trees remain positive definite whenever the input matrix is positive definite. We demonstrate that the resulting algorithm is closely related to other iterative schemes for quadratic minimization such as the Gauss-Seidel and Jacobi algorithms. Finally, we observe, empirically, that there always exists a choice of parameters such that the above generalization of the GaBP algorithm converges.

The max-product algorithm, a local message-passing scheme that attempts to compute the most probable assignment (MAP) of a given probability distribution, has been successfully employed as a method of approximate inference for applications arising in coding theory, computer vision, and machine learning. However, the max-product algorithm is not guaranteed to converge to the MAP assignment, and if it does, is not guaranteed to recover the MAP assignment. Alternative convergent message-passing schemes have been proposed to overcome these difficulties. This work provides a systematic study of such message-passing algorithms that extends the known results by exhibiting new sufficient conditions for convergence to local and/or global optima, providing a combinatorial characterization of these optima based on graph covers, and describing a new convergent and correct message-passing algorithm whose derivation unifies many of the known convergent message-passing algorithms. While convergent and correct message-passing algorithms represent a step forward in the analysis of max-product style message-passing algorithms, the conditions needed to guarantee convergence to a global optimum can be too restrictive in both theory and practice. This limitation of convergent and correct message-passing schemes is characterized by graph covers and illustrated by example.

In this paper we propose an extension of Defeasible Logic to represent and compute three concepts of defeasible permission. In particular, we discuss different types of explicit permissive norms that work as exceptions to opposite obligations. Moreover, we show how strong permissions can be represented both with, and without introducing a new consequence relation for inferring conclusions from explicit permissive norms. Finally, we illustrate how a preference operator applicable to contrary-to-duty obligations can be combined with a new operator representing ordered sequences of strong permissions which derogate from prohibitions. The logical system is studied from a computational standpoint and is shown to have liner computational complexity. The concept of permission plays an important role in many normative domains in that it may be crucial in characterising notions such as those of authorisation and derogation [11,30,33]. For example, sometimes it may happen that we mistakenly drive to a building site, or a roadwork restricted area, with signs out saying "No admittance.

The optimal selection of experimental conditions is essential to maximizing the value of data for inference and prediction, particularly in situations where experiments are time-consuming and expensive to conduct. We propose a general mathematical framework and an algorithmic approach for optimal experimental design with nonlinear simulation-based models; in particular, we focus on finding sets of experiments that provide the most information about targeted sets of parameters. Our framework employs a Bayesian statistical setting, which provides a foundation for inference from noisy, indirect, and incomplete data, and a natural mechanism for incorporating heterogeneous sources of information. An objective function is constructed from information theoretic measures, reflecting expected information gain from proposed combinations of experiments. Polynomial chaos approximations and a two-stage Monte Carlo sampling method are used to evaluate the expected information gain. Stochastic approximation algorithms are then used to make optimization feasible in computationally intensive and high-dimensional settings. These algorithms are demonstrated on model problems and on nonlinear parameter estimation problems arising in detailed combustion kinetics.

We analyse the computational complexity of three problems in judgment aggregation: (1) computing a collective judgment from a profile of individual judgments (the winner determination problem); (2) deciding whether a given agent can influence the outcome of a judgment aggregation procedure in her favour by reporting insincere judgments (the strategic manipulation problem); and (3) deciding whether a given judgment aggregation scenario is guaranteed to result in a logically consistent outcome, independently from what the judgments supplied by the individuals are (the problem of the safety of the agenda). We provide results both for specific aggregation procedures (the quota rules, the premise-based procedure, and a distance-based procedure) and for classes of aggregation procedures characterised in terms of fundamental axioms.

We present an algorithm, AROFAC2, which detects the (CP-)rank of a degree 3 tensor and calculates its factorization into rank-one components. We provide generative conditions for the algorithm to work and demonstrate on both synthetic and real world data that AROFAC2 is a potentially outperforming alternative to the gold standard PARAFAC over which it has the advantages that it can intrinsically detect the true rank, avoids spurious components, and is stable with respect to outliers and non-Gaussian noise.

In this paper, a new method is proposed for sparse PCA based on the recursive divide-and-conquer methodology. The main idea is to separate the original sparse PCA problem into a series of much simpler sub-problems, each having a closed-form solution. By recursively solving these sub-problems in an analytical way, an efficient algorithm is constructed to solve the sparse PCA problem. The algorithm only involves simple computations and is thus easy to implement. The proposed method can also be very easily extended to other sparse PCA problems with certain constraints, such as the nonnegative sparse PCA problem. Furthermore, we have shown that the proposed algorithm converges to a stationary point of the problem, and its computational complexity is approximately linear in both data size and dimensionality. The effectiveness of the proposed method is substantiated by extensive experiments implemented on a series of synthetic and real data in both reconstruction-error-minimization and data-variance-maximization viewpoints.

In game theory, an Evolutionarily Stable Set (ES set) is a set of Nash Equilibrium (NE) strategies that give the same payoffs. Similar to an Evolutionarily Stable Strategy (ES strategy), an ES set is also a strict NE. This work investigates the evolutionary stability of classical and quantum strategies in the quantum penny flip games. In particular, we developed an evolutionary game theory model to conduct a series of simulations where a population of mixed classical strategies from the ES set of the game were invaded by quantum strategies. We found that when only one of the two players' mixed classical strategies were invaded, the results were different. In one case, due to the interference phenomenon of superposition, quantum strategies provided more payoff, hence successfully replaced the mixed classical strategies in the ES set. In the other case, the mixed classical strategies were able to sustain the invasion of quantum strategies and remained in the ES set. Moreover, when both players' mixed classical strategies were invaded by quantum strategies, a new quantum ES set emerged. The strategies in the quantum ES set give both players payoff 0, which is the same as the payoff of the strategies in the mixed classical ES set of this game.

Producing overlapping schemes is a major issue in clustering. Recent proposed overlapping methods relies on the search of an optimal covering and are based on different metrics, such as Euclidean distance and I-Divergence, used to measure closeness between observations. In this paper, we propose the use of another measure for overlapping clustering based on a kernel similarity metric .We also estimate the number of overlapped clusters using the Gram matrix. Experiments on both Iris and EachMovie datasets show the correctness of the estimation of number of clusters and show that measure based on kernel similarity metric improves the precision, recall and f-measure in overlapping clustering.

Overlapping clustering problem is an important learning issue in which clusters are not mutually exclusive and each object may belongs simultaneously to several clusters. This paper presents a kernel based method that produces overlapping clusters on a high feature space using mercer kernel techniques to improve separability of input patterns. The proposed method, called OKM-K(Overlapping $k$-means based kernel method), extends OKM (Overlapping $k$-means) method to produce overlapping schemes. Experiments are performed on overlapping dataset and empirical results obtained with OKM-K outperform results obtained with OKM.