A central problem in applied data analysis is time series In this paper, we develop a tractable approach to maximum modeling--estimating and forecasting a discrete-time likelihood parameter estimation for stochastic multivariate stochastic process--for which the autoregressive moving ARMA models. To efficiently compute a globally average (ARMA) and stochastic ARMA (Thiesson et al. optimal estimate, the problem is re-expressed as a regularized 2012) are fundamental models. An ARMA model describes loss minimization, which then allows recent algorithmic the behavior of a linear dynamical system under advances in sparse estimation to be applied (Shah et al. latent Gaussian perturbations (Brockwell and Davis 2002; 2012; Candes et al. 2011; Bach, Mairal, and Ponce 2008; Lütkepohl 2007), which affords intuitive modeling capability, Zhang et al. 2011; White et al. 2012). Although there has efficient forecasting algorithms, and a close relationship been recent progress in global estimation for ARMA, such to linear Gaussian state-space models (Katayama 2006, approaches have either been restricted to single-input singleoutput pp.5-6).

In this paper, we propose an interpretation of active learning from a pure algebraic view and combine it with semi-supervised manifold learning. The proposed active manifold learning algorithm aims to learn the low-dimensional parameter space of the manifold with high accuracy from smartly labeled samples. We demonstrate that this problem is equivalent to a condition number minimization problem of the alignment matrix. Focusing on this problem, we first give a theoretical upper bound for the solution. Then we develop a heuristic but effective sample selection algorithm with the help of the Gershgorin circle theorem. We investigate the rationality, the feasibility, the universality and the complexity of the proposed method and demonstrate that our method yields encouraging active learning results.

In many learning tasks with structural properties, structural sparsity methods help induce sparse models, usually leading to better interpretability and higher generalization performance. One popular approach is to use group sparsity regularization that enforces sparsity on the clustered groups of features, while another popular approach is to adopt graph sparsity regularization that considers sparsity on the link structure of graph embedded features. Both the group and graph structural properties co-exist in many applications. However, group sparsity and graph sparsity have not been considered simultaneously yet. In this paper, we propose a g 2 -regularization that takes group and graph sparsity into joint consideration, and present an effective approach for its optimization. Experiments on both synthetic and real data show that, enforcing group-graph sparsity lead to better performance than using group sparsity or graph sparsity only.

Word alignment is an important natural language processing task that indicates the correspondence between natural languages. Recently, unsupervised learning of log-linear models for word alignment has received considerable attention as it combines the merits of generative and discriminative approaches. However, a major challenge still remains: it is intractable to calculate the expectations of non-local features that are critical for capturing the divergence between natural languages. We propose a contrastive approach that aims to differentiate observed training examples from noises. It not only introduces prior knowledge to guide unsupervised learning but also cancels out partition functions. Based on the observation that the probability mass of log-linear models for word alignment is usually highly concentrated, we propose to use top-$n$ alignments to approximate the expectations with respect to posterior distributions. This allows for efficient and accurate calculation of expectations of non-local features. Experiments show that our approach achieves significant improvements over state-of-the-art unsupervised word alignment methods.

The selection of kernel function which determines the mapping between the input space and the feature space is of crucial importance to kernel methods. Existing kernel selection approaches commonly use some measures of generalization error, which are usually difficult to estimate and have slow convergence rates. In this paper, we propose a novel measure, called eigenvalues ratio (ER), of the tight bound of generalization error for kernel selection. ER is the ration between the sum of the main eigenvalues and that of the tail eigenvalues of the kernel matrix. Defferent from most of existing measures, ER is defined on the kernel matrxi, so it can be estimated easily from the available training data, which makes it usable for kernel selection. We establish tight ER-based generalization error bounds of order $O(\frac{1}{n})$ for several kernel-based methods under certain general conditions, while for most of existing measures, the convergence rate is at most $O(\frac{1}{\sqrt{n}})$. Finally, to guarantee good generalization performance, we propose a novel kernel selection criterion by minimizing the derived tight generalization error bounds. Theoretical analysis and experimental results demonstrate that our kernel selection criterion is a good choice for kernel seletion.

Group sparsity has drawn much attention in machine learning. However, existing work can handle only datasets with certain group structures, where each sample has a certain membership with one or more groups. This paper investigates the learning of sparse representations from datasets with uncertain group structures, where each sample has an uncertain member-ship with all groups in terms of a probability distribution. We call this problem uncertain group sparse representation (UGSR in short), which is a generalization of the standard group sparse representation (GSR). We formulate the UGSR model and propose an efficient algorithm to solve this problem. We apply UGSR to text emotion classification and aging face recognition. Experiments show that UGSR outperforms standard sparse representation (SR) and standard GSR as well as fuzzy kNN classification.

Supervised dimensionality reduction has shown great advantages in finding predictive subspaces. Previous methods rarely consider the popular maximum margin principle and are prone to overfitting to usually small training data, especially for those under the maximum likelihood framework. In this paper, we present a posterior-regularized Bayesian approach to combine Principal Component Analysis (PCA) with the max-margin learning. Based on the data augmentation idea for max-margin learning and the probabilistic interpretation of PCA, our method can automatically infer the weight and penalty parameter of max-margin learning machine, while finding the most appropriate PCA subspace simultaneously under the Bayesian framework. We develop a fast mean-field variational inference algorithm to approximate the posterior. Experimental results on various classification tasks show that our method outperforms a number of competitors.

Interactions among people or objects are often dynamic in nature and can be represented as a sequence of networks, each providing a snapshot of the interactions over a brief period of time. An important task in analyzing such evolving networks is change-point detection, in which we both identify the times at which the large-scale pattern of interactions changes fundamentally and quantify how large and what kind of change occurred. Here, we formalize for the first time the network change-point detection problem within an online probabilistic learning framework and introduce a method that can reliably solve it. This method combines a generalized hierarchical random graph model with a Bayesian hypothesis test to quantitatively determine if, when, and precisely how a change point has occurred. We analyze the detectability of our method using synthetic data with known change points of different types and magnitudes, and show that this method is more accurate than several previously used alternatives. Applied to two high-resolution evolving social networks, this method identifies a sequence of change points that align with known external ``shocks'' to these networks.

Conditional Value at Risk (CVaR) is a prominent risk measure that is being used extensively in various domains. We develop a new formula for the gradient of the CVaR in the form of a conditional expectation. Based on this formula, we propose a novel sampling-based estimator for the gradient of the CVaR, in the spirit of the likelihood-ratio method. We analyze the bias of the estimator, and prove the convergence of a corresponding stochastic gradient descent algorithm to a local CVaR optimum. Our method allows to consider CVaR optimization in new domains. As an example, we consider a reinforcement learning application, and learn a risk-sensitive controller for the game of Tetris.

Current manifold alignment methods can effectively align data sets that are drawn from a non-intersecting set of manifolds. However, as data sets become increasingly high-dimensional and complex, this assumption may not hold. This paper proposes a novel manifold alignment algorithm, low rank alignment (LRA), that uses a low rank representation (instead of a nearest neighbor graph construction) to embed and align data sets drawn from mixtures of manifolds. LRA does not require the tuning of a sensitive nearest neighbor hyperparameter or prior knowledge of the number of manifolds, both of which are common drawbacks with existing techniques. We demonstrate the effectiveness of our algorithm in two real-world applications: a transfer learning task in spectroscopy and a canonical information retrieval task.