--CANDECOMP/P ARAFAC (CP) tensor factorization of incomplete data is a powerful technique for tensor completion through explicitly capturing the multilinear latent factors. The existing CP algorithms require the tensor rank to be manually specified, however, the determination of tensor rank remains a challenging problem especially for CP rank . In addition, existing approaches do not take into account uncertainty information of latent factors, as well as missing entries. T o address these issues, we formulate CP factorization using a hierarchical probabilistic model and employ a fully Bayesian treatment by incorporating a sparsity-inducing prior over multiple latent factors and the appropriate hyperpriors over all hyperparameters, resulting in automatic rank determination. T o learn the model, we develop an efficient deterministic Bayesian inference algorithm, which scales linearly with data size. Our method is characterized as a tuning parameter-free approach, which can effectively infer underlying multilinear factors with a low-rank constraint, while also providing predictive distributions over missing entries. Extensive simulations on synthetic data illustrate the intrinsic capability of our method to recover the ground-truth of CP rank and prevent the overfitting problem, even when a large amount of entries are missing. Moreover, the results from real-world applications, including image inpainting and facial image synthesis, demonstrate that our method outperforms state-of-the-art approaches for both tensor factorization and tensor completion in terms of predictive performance. For instance, a video sequence can be represented by a third-order tensor with dimensionality of height width time; an image ensemble measured under multiple conditions can be represented by a higher order tensor with dimensionality ofpixel person pose illumination . T ensor factorization enables us to explicitly take into account the structure information by effectively capturing the multilinear interactions among multiple latent factors. Therefore, its theory and algorithms have been an active area of study during the past decade (see e.g., [1], [2]), and have been successfully applied to various application fields, such as face recognition, social network analysis, image and video completion, and brain signal processing. This issue has attracted a great deal of research interest in tensor completion in recent years. The objective of tensor factorization of incomplete data is to capture the underlying multilinear factors from only partially observed entries, which can in turn predict the missing entries.

Inference in general Ising models is difficult, due to high treewidth making tree-based algorithms intractable. Moreover, when interactions are strong, Gibbs sampling may take exponential time to converge to the stationary distribution. We present an algorithm to project Ising model parameters onto a parameter set that is guaranteed to be fast mixing, under several divergences. We find that Gibbs sampling using the projected parameters is more accurate than with the original parameters when interaction strengths are strong and when limited time is available for sampling.

In this manuscript we consider the problem of jointly estimating multiple graphical models in high dimensions. We assume that the data are collected from n subjects, each of which consists of T possibly dependent observations. The graphical models of subjects vary, but are assumed to change smoothly corresponding to a measure of closeness between subjects. We propose a kernel based method for jointly estimating all graphical models. Theoretically, under a double asymptotic framework, where both (T,n) and the dimension d can increase, we provide the explicit rate of convergence in parameter estimation. It characterizes the strength one can borrow across different individuals and impact of data dependence on parameter estimation. Empirically, experiments on both synthetic and real resting state functional magnetic resonance imaging (rs-fMRI) data illustrate the effectiveness of the proposed method.

While most Bayesian nonparametric models in machine learning have focused on the Dirichlet process, the beta process, or their variants, the gamma process has recently emerged as a useful nonparametric prior in its own right. Current inference schemes for models involving the gamma process are restricted to MCMC-based methods, which limits their scalability. In this paper, we present a variational inference framework for models involving gamma process priors. Our approach is based on a novel stick-breaking constructive definition of the gamma process. We prove correctness of this stick-breaking process by using the characterization of the gamma process as a completely random measure (CRM), and we explicitly derive the rate measure of our construction using Poisson process machinery. We also derive error bounds on the truncation of the infinite process required for variational inference, similar to the truncation analyses for other nonparametric models based on the Dirichlet and beta processes. Our representation is then used to derive a variational inference algorithm for a particular Bayesian nonparametric latent structure formulation known as the infinite Gamma-Poisson model, where the latent variables are drawn from a gamma process prior with Poisson likelihoods. Finally, we present results for our algorithms on nonnegative matrix factorization tasks on document corpora, and show that we compare favorably to both sampling-based techniques and variational approaches based on beta-Bernoulli priors.

Restricted Boltzmann Machines (RBMs) are one of the fundamental building blocks of deep learning. Approximate maximum likelihood training of RBMs typically necessitates sampling from these models. In many training scenarios, computationally efficient Gibbs sampling procedures are crippled by poor mixing. In this work we propose a novel method of sampling from Boltzmann machines that demonstrates a computationally efficient way to promote mixing. Our approach leverages an under-appreciated property of deep generative models such as the Deep Belief Network (DBN), where Gibbs sampling from deeper levels of the latent variable hierarchy results in dramatically increased ergodicity. Our approach is thus to train an auxiliary latent hierarchical model, based on the DBN. When used in conjunction with parallel-tempering, the method is asymptotically guaranteed to simulate samples from the target RBM. Experimental results confirm the effectiveness of this sampling strategy in the context of RBM training.

Probabilistic approaches for tensor factorization aim to extract meaningful structure from incomplete data by postulating low rank constraints. Recently, variational Bayesian (VB) inference techniques have successfully been applied to large scale models. This paper presents full Bayesian inference via VB on both single and coupled tensor factorization models. Our method can be run even for very large models and is easily implemented. It exhibits better prediction performance than existing approaches based on maximum likelihood on several real-world datasets for missing link prediction problem.

Concept map is a graphical tool for representing knowledge. They have been used in many different areas, including education, knowledge management, business and intelligence. Constructing of concept maps manually can be a complex task; an unskilled person may encounter difficulties in determining and positioning concepts relevant to the problem area. An application that recommends concept candidates and their position in a concept map can significantly help the user in that situation. This paper gives an overview of different approaches to automatic and semi-automatic creation of concept maps from textual and non-textual sources. The concept map mining process is defined, and one method suitable for the creation of concept maps from unstructured textual sources in highly inflected languages such as the Croatian language is described in detail. Proposed method uses statistical and data mining techniques enriched with linguistic tools. With minor adjustments, that method can also be used for concept map mining from textual sources in other morphologically rich languages.

In (exploratory) factor analysis, the loading matrix is identified only up to orthogonal rotation. For identifiability, one thus often takes the loading matrix to be lower triangular with positive diagonal entries. In Bayesian inference, a standard practice is then to specify a prior under which the loadings are independent, the off-diagonal loadings are normally distributed, and the diagonal loadings follow a truncated normal distribution. This prior specification, however, depends in an important way on how the variables and associated rows of the loading matrix are ordered. We show how a minor modification of the approach allows one to compute with the identifiable lower triangular loading matrix but maintain invariance properties under reordering of the variables.

Maximum likelihood is the most widely used statistical estimation technique. Recent work by Jiao, Venkat, Han, and Weissman [1] introduced a general methodology for the construction of estimators for functionals in parametric models, and demonstrated improvements - both in theory and in practice - over the maximum likelihood estimator (MLE), particularly in high dimensional scenarios involving parameter dimension comparable to or larger than the number of samples. This approach to estimation, building on results from approximation theory, is shown to yield minimax rate-optimal estimators for a wide class of functionals, implementable with modest computational requirements. In a nutshell, a message of this recent work is that, for a wide class of functionals, the performance of these essentially optimal estimators with n samples is comparable to that of the MLE with nlnn samples. In the present paper, we highlight the applicability of the aforementioned methodology to statistical problems beyond functional estimation, and show that it can yield substantial gains. For example, we demonstrate that for learning tree-structured graphical models, our approach achieves a significant reduction of the required data size compared with the classical Chow-Liu algorithm, which is an implementation of the MLE, to achieve the same accuracy. The key step in improving the Chow-Liu algorithm is to replace the empirical mutual information with the estimator for mutual information proposed in [1]. Further, applying the same replacement approach to classical Bayesian network classification, the resulting classifiers uniformly outperform the previous classifiers on 26 widely used datasets.

Jump Markov linear models consists of a finite number of linear state space models and a discrete variable encoding the jumps (or switches) between the different linear models. Identifying jump Markov linear models makes for a challenging problem lacking an analytical solution. We derive a new expectation maximization (EM) type algorithm that produce maximum likelihood estimates of the model parameters. Our development hinges upon recent progress in combining particle filters with Markov chain Monte Carlo methods in solving the nonlinear state smoothing problem inherent in the EM formulation. Key to our development is that we exploit a conditionally linear Gaussian substructure in the model, allowing for an efficient algorithm.