In this paper we consider sparse approximation problems, that is, general $l_0$ minimization problems with the $l_0$-"norm" of a vector being a part of constraints or objective function. In particular, we first study the first-order optimality conditions for these problems. We then propose penalty decomposition (PD) methods for solving them in which a sequence of penalty subproblems are solved by a block coordinate descent (BCD) method. Under some suitable assumptions, we establish that any accumulation point of the sequence generated by the PD methods satisfies the first-order optimality conditions of the problems. Furthermore, for the problems in which the $l_0$ part is the only nonconvex part, we show that such an accumulation point is a local minimizer of the problems. In addition, we show that any accumulation point of the sequence generated by the BCD method is a saddle point of the penalty subproblem. Moreover, for the problems in which the $l_0$ part is the only nonconvex part, we establish that such an accumulation point is a local minimizer of the penalty subproblem. Finally, we test the performance of our PD methods by applying them to sparse logistic regression, sparse inverse covariance selection, and compressed sensing problems. The computational results demonstrate that our methods generally outperform the existing methods in terms of solution quality and/or speed.

Over the past decades, statisticians and machine-learning researchers have developed literally thousands of new tools for the reduction of high-dimensional data in order to identify the variables most responsible for a particular trait. These tools have applications in a plethora of settings, including data analysis in the fields of business, education, forensics, and biology (such as microarray, proteomics, brain imaging), to name a few. In the present work, we focus our investigation on the limitations and potential misuses of certain tools in the analysis of the benchmark colon cancer data (2,000 variables; Alon et al., 1999) and the prostate cancer data (6,033 variables; Efron, 2010, 2008). Our analysis demonstrates that models that produce 100% accuracy measures often select different sets of genes and cannot stand the scrutiny of parameter estimates and model stability. Furthermore, we created a host of simulation datasets and "artificial diseases" to evaluate the reliability of commonly used statistical and data mining tools. We found that certain widely used models can classify the data with 100% accuracy without using any of the variables responsible for the disease. With moderate sample size and suitable pre-screening, stochastic gradient boosting will be shown to be a superior model for gene selection and variable screening from high-dimensional datasets.

Cardinality constraints or, more generally, weight constraints are well recognized as an important extension of answer-set programming. Clearly, all common algorithmic tasks related to programs with cardinality or weight constraints - like checking the consistency of a program - are intractable. Many intractable problems in the area of knowledge representation and reasoning have been shown to become linear time tractable if the treewidth of the programs or formulas under consideration is bounded by some constant. The goal of this paper is to apply the notion of treewidth to programs with cardinality or weight constraints and to identify tractable fragments. It will turn out that the straightforward application of treewidth to such class of programs does not suffice to obtain tractability. However, by imposing further restrictions, tractability can be achieved.

This paper addresses a mixed integer programming (MIP) formulation for the multi-item uncapacitated lot-sizing problem that is inspired from the trailer manufacturer. The proposed MIP model has been utilized to find out the optimum order quantity, optimum order time, and the minimum total cost of purchasing, ordering, and holding over the predefined planning horizon. This problem is known as NP-hard problem. The model was presented in an optimal software form using LINGO 13.0.

This paper presents theory for Normalized Random Measures (NRMs), Normalized Generalized Gammas (NGGs), a particular kind of NRM, and Dependent Hierarchical NRMs which allow networks of dependent NRMs to be analysed. These have been used, for instance, for time-dependent topic modelling. In this paper, we first introduce some mathematical background of completely random measures (CRMs) and their construction from Poisson processes, and then introduce NRMs and NGGs. Slice sampling is also introduced for posterior inference. The dependency operators in Poisson processes and for the corresponding CRMs and NRMs is then introduced and Posterior inference for the NGG presented. Finally, we give dependency and composition results when applying these operators to NRMs so they can be used in a network with hierarchical and dependent relations.

High throughput genetic sequencing arrays with thousands of measurements per sample and a great amount of related censored clinical data have increased demanding need for better measurement specific model selection. In this paper we establish strong oracle properties of nonconcave penalized methods for nonpolynomial (NP) dimensional data with censoring in the framework of Cox's proportional hazards model. A class of folded-concave penalties are employed and both LASSO and SCAD are discussed specifically. We unveil the question under which dimensionality and correlation restrictions can an oracle estimator be constructed and grasped. It is demonstrated that nonconcave penalties lead to significant reduction of the "irrepresentable condition" needed for LASSO model selection consistency. The large deviation result for martingales, bearing interests of its own, is developed for characterizing the strong oracle property. Moreover, the nonconcave regularized estimator, is shown to achieve asymptotically the information bound of the oracle estimator. A coordinate-wise algorithm is developed for finding the grid of solution paths for penalized hazard regression problems, and its performance is evaluated on simulated and gene association study examples.

The sum-product or belief propagation (BP) algorithm is a widely-used message-passing algorithm for computing marginal distributions in graphical models with discrete variables. At the core of the BP message updates, when applied to a graphical model with pairwise interactions, lies a matrix-vector product with complexity that is quadratic in the state dimension $d$, and requires transmission of a $(d-1)$-dimensional vector of real numbers (messages) to its neighbors. Since various applications involve very large state dimensions, such computation and communication complexities can be prohibitively complex. In this paper, we propose a low-complexity variant of BP, referred to as stochastic belief propagation (SBP). As suggested by the name, it is an adaptively randomized version of the BP message updates in which each node passes randomly chosen information to each of its neighbors. The SBP message updates reduce the computational complexity (per iteration) from quadratic to linear in $d$, without assuming any particular structure of the potentials, and also reduce the communication complexity significantly, requiring only $\log{d}$ bits transmission per edge. Moreover, we establish a number of theoretical guarantees for the performance of SBP, showing that it converges almost surely to the BP fixed point for any tree-structured graph, and for graphs with cycles satisfying a contractivity condition. In addition, for these graphical models, we provide non-asymptotic upper bounds on the convergence rate, showing that the $\ell_{\infty}$ norm of the error vector decays no slower than $O(1/\sqrt{t})$ with the number of iterations $t$ on trees and the mean square error decays as $O(1/t)$ for general graphs. These analysis show that SBP can provably yield reductions in computational and communication complexities for various classes of graphical models.

The probabilistic graphs framework models the uncertainty inherent in real-world domains by means of probabilistic edges whose value quantifies the likelihood of the edge existence or the strength of the link it represents. The goal of this paper is to provide a learning method to compute the most likely relationship between two nodes in a framework based on probabilistic graphs. In particular, given a probabilistic graph we adopted the language-constraint reachability method to compute the probability of possible interconnections that may exists between two nodes. Each of these connections may be viewed as feature, or a factor, between the two nodes and the corresponding probability as its weight. Each observed link is considered as a positive instance for its corresponding link label. Given the training set of observed links a L2-regularized Logistic Regression has been adopted to learn a model able to predict unobserved link labels. The experiments on a real world collaborative filtering problem proved that the proposed approach achieves better results than that obtained adopting classical methods.

We provide a unifying framework linking two classes of statistics used in two-sample and independence testing: on the one hand, the energy distances and distance covariances from the statistics literature; on the other, distances between embeddings of distributions to reproducing kernel Hilbert spaces (RKHS), as established in machine learning. The equivalence holds when energy distances are computed with semimetrics of negative type, in which case a kernel may be defined such that the RKHS distance between distributions corresponds exactly to the energy distance. We determine the class of probability distributions for which kernels induced by semimetrics are characteristic (that is, for which embeddings of the distributions to an RKHS are injective). Finally, we investigate the performance of this family of kernels in two-sample and independence tests: we show in particular that the energy distance most commonly employed in statistics is just one member of a parametric family of kernels, and that other choices from this family can yield more powerful tests.

We develop the Latent Multi-group Membership Graph (LMMG) model, a model of networks with rich node feature structure. In the LMMG model, each node belongs to multiple groups and each latent group models the occurrence of links as well as the node feature structure. The LMMG can be used to summarize the network structure, to predict links between the nodes, and to predict missing features of a node. We derive efficient inference and learning algorithms and evaluate the predictive performance of the LMMG on several social and document network datasets.