Estimation of population covariance matrices from samples of multivariate data has draw many attentions in the last decade owing to its fundamental importance in multivariate analysis. With dramatic advances in technology in recent years, various research fields, such as genetic data, brain imaging, spectroscopic imaging, climate data and so on, have been used to deal with massive highdimensional data sets, whose sample sizes can be very small relative to dimension. In such settings, the standard and the most usual sample covariance matrices often performs poorly [1, 2, 11]. Fortunately, regularization as a class of new methods to estimate covariance matrices has recently emerged to overcome those shortages of using traditional sample covariance matrices. These methods encompass several specified forms, banding [1, 6, 17], tapering [4, 10] and thresholding [2, 5, 8, 16] for instance.

Restricted Boltzmann Machines (RBM) [9, 5] have recently attracted an increasing attention for their rich capacity in a variety of learning tasks, including multivariate distribution modelling, feature extraction, classification, and construction of deep architectures [8, 19]. An RBM is a two-layer Markov random field in which the visible layer represents observed variables and the hidden layer represents latent aspects of the data. Pairwise interactions are only permitted for units between layers. As a result, the posterior distribution over the hidden variables and the probability of the data generative model are easy to evaluate, allowing fast feature extraction and efficient sampling-based inference [7]. Nonetheless, most existing work in RBMs implicitly assumes that the visible layer contains variables of the same modality. By far the most popular input types are binary [5] and Gaussian [8]. Recent extension includes categorical [21], ordinal [25], Poisson [6] and Beta [13] data. To the best of our knowledge, none has been considered for multicategorical and category-ranking data, nor for a mixed combination of these data types. In this paper, we investigate a generalisation of the RBM for variables of multiple modalities and types.

Restricted Boltzmann machines (RBMs) have proved to be a versatile tool for a wide variety of machine learning tasks and as a building block for deep architectures [12, 24, 28]. The original proposals mainly handle binary visible and hidden units. Whilst binary hidden units are broadly applicable as feature detectors, non-binary visible data requires different designs. Recent extensions to other data types result in type-dependent models: the Gaussian for continuous inputs [12], Beta for bounded continuous inputs [16], Poisson for count data [9], multinomial for unordered categories [25], and ordinal models for ordered categories [37, 35]. The Boltzmann distribution permits several types to be jointly modelled, thus making the RBM a good tool for multimodal and complex social survey analysis. The work of [20, 29, 40] combines continuous (e.g., visual and audio) and discrete modalities (e.g., words). The work of [34] extends the idea further to incorporate ordinal and rank data. However, there are conceptual drawbacks: First, conditioned on the hidden layer, they are still separate type-specific models; second, handling ordered categories and ranks is not natural; and third, specifying direct correlation between these types remains difficult. The main thesis of this paper is that many data types can be captured in one unified model.

In Bayesian analysis the form of the posterior distribution is often not analytically tractable. To obtain quantities of interest under such a distribution, such as moments or marginal distributions, we typically need to use Monte Carlo methods or approximate the posterior with a more convenient distribution. A popular method of obtaining such an approximation is structured or fixed-form Variational Bayes, which works by numerically minimizing the Kullback-Leibler divergence of an approximating distribution in the exponential family to the intractable target distribution (Attias, 2000; Beal and Ghahramani, 2006; Jordan et al., 1999; Wainwright and Jordan, 2008). For certain problems, algorithms exist that can solve this optimization problem in much less time than it would take to approximate the posterior using Monte Carlo methods (see e.g.

It is well known that good initializations can improve the speed and accuracy of the solutions of many nonnegative matrix factorization (NMF) algorithms. Many NMF algorithms are sensitive with respect to the initialization of W or H or both. This is especially true of algorithms of the alternating least squares (ALS) type, including the two new ALS algorithms that we present in this paper. We compare the results of six initialization procedures (two standard and four new) on our ALS algorithms. Lastly, we discuss the practical issue of choosing an appropriate convergence criterion.

This paper considers the problem of clustering a partially observed unweighted graph---i.e., one where for some node pairs we know there is an edge between them, for some others we know there is no edge, and for the remaining we do not know whether or not there is an edge. We want to organize the nodes into disjoint clusters so that there is relatively dense (observed) connectivity within clusters, and sparse across clusters. We take a novel yet natural approach to this problem, by focusing on finding the clustering that minimizes the number of "disagreements"---i.e., the sum of the number of (observed) missing edges within clusters, and (observed) present edges across clusters. Our algorithm uses convex optimization; its basis is a reduction of disagreement minimization to the problem of recovering an (unknown) low-rank matrix and an (unknown) sparse matrix from their partially observed sum. We evaluate the performance of our algorithm on the classical Planted Partition/Stochastic Block Model. Our main theorem provides sufficient conditions for the success of our algorithm as a function of the minimum cluster size, edge density and observation probability; in particular, the results characterize the tradeoff between the observation probability and the edge density gap. When there are a constant number of clusters of equal size, our results are optimal up to logarithmic factors.

False-name manipulation refers to the question of whether a player in a weighted voting game can increase her power by splitting into several players and distributing her weight among these false identities. Relatedly, the beneficial merging problem asks whether a coalition of players can increase their power in a weighted voting game by merging their weights. For the problems of whether merging or splitting players in weighted voting games is beneficial in terms of the Shapley--Shubik and the normalized Banzhaf index, merely NP-hardness lower bounds are known, leaving the question about their exact complexity open. For the Shapley--Shubik and the probabilistic Banzhaf index, we raise these lower bounds to hardness for PP, "probabilistic polynomial time," a class considered to be by far a larger class than NP. For both power indices, we provide matching upper bounds for beneficial merging and, whenever the new players' weights are given, also for beneficial splitting, thus resolving previous conjectures in the affirmative. Relatedly, we consider the beneficial annexation problem, asking whether a single player can increase her power by taking over other players' weights. It is known that annexation is never disadvantageous for the Shapley--Shubik index, and that beneficial annexation is NP-hard for the normalized Banzhaf index. We show that annexation is never disadvantageous for the probabilistic Banzhaf index either, and for both the Shapley--Shubik index and the probabilistic Banzhaf index we show that it is NP-complete to decide whether annexing another player is advantageous. Moreover, we propose a general framework for merging and splitting that can be applied to different classes and representations of games.

Improving public health is a major responsibility of any government, and is of major interest to citizens and scientific communities around the world. Here, one sees two extremes. On one hand, tremendous progress has been made in recent years in the understanding of causes, spread and remedies of common and regularly occurring diseases like Dengue, Malaria and Japanese Encephalistis (JE). On the other hand, public agencies treat these diseases in an ad hoc manner without learning from the experiences of previous years. Specifically, they would get alerted once reported cases have already arisen substantially in the known disease season, reactively initiate a few actions and then document the disease impact (cases, deaths) for that period, only to forget this learning in the next season. However, they miss the opportunity to reduce preventable deaths and sickness, and their corresponding economic impact, which scientific progress could have enabled. The gap is universal but very prominent in developing countries like India.   In this paper, we show that if public agencies provide historical disease impact information openly, it can be analyzed with statistical and machine learning techniques, correlated with best emerging practices in disease control, and simulated in a setting to optimize social benefits to provide timely guidance for new disease seasons and regions. We illustrate using open data for mosquito-borne communicable diseases; published results in public health on efficacy of Dengue control methods and apply it on a simulated typical city for maximal benefits with available resources. The exercise helps us further suggest strategies for new regions that may be anywhere in the world, how data could be better recorded by city agencies and what prevention methods should medical community focus on for wider impact.

Conventional urban traffic control systems have been based on historical traffic data. Later advancements made use of detectors, which enabled the gathering of real time traffic data, in order to re organize and calibrate traffic signalization programs. Further evolvement provided the ability to forecast traffic conditions, in order to develop traffic signalization programs and strategies pre computed and applied at the most appropriate time frame for the optimal control of the current traffic conditions. We, propose the next generation of traffic control systems based on principles of Artificial Intelligence and Context Awareness. Most of the existing algorithms use average waiting time or length of the queue to assess an algorithm's performance. However, a low average waiting time may come at the cost of delaying other vehicles indefinitely. In our algorithm, besides the vehicle queue, we use'fairness' also as an important performance metric to assess an algorithm's performance.

Optimization problems arise in various disciplines such as engineering design, manufacturing system, economics etc. thus in view of the practical utility of optimization problems there is a need for efficient and robust computational algorithms which can solve optimization problems arising in different fields. Several NPhard combinatorial optimization problems, such as the traveling salesman problem, and yard management of container terminals can be modeled as QAPs.. Optimization is a process that finds a best, or optimal, solution for a problem. An optimization problem is defined as: Finding values of the variables that minimize or maximize the objective function while satisfying the constraints. The Optimization problems are centered on three factors: (1) an objective function which is to be minimized or maximized.