We consider the class of optimization problems arising from computationally intensive L1-regularized M-estimators, where the function or gradient values are very expensive to compute. A particular instance of interest is the L1-regularized MLE for learning Conditional Random Fields (CRFs), which are a popular class of statistical models for varied structured prediction problems such as sequence labeling, alignment, and classification with label taxonomy. L1-regularized MLEs for CRFs are particularly expensive to optimize since computing the gradient values requires an expensive inference step. In this work, we propose the use of a carefully constructed proximal quasi-Newton algorithm for such computationally intensive M-estimation problems, where we employ an aggressive active set selection technique. In a key contribution of the paper, we show that the proximal quasi-Newton method is provably super-linearly convergent, even in the absence of strong convexity, by leveraging a restricted variant of strong convexity. In our experiments, the proposed algorithm converges considerably faster than current state-of-the-art on the problems of sequence labeling and hierarchical classification.

We develop latent variable models for Bayesian learning based low-rank matrix completion and reconstruction from linear measurements. For under-determined systems, the developed methods are shown to reconstruct low-rank matrices when neither the rank nor the noise power is known a-priori. We derive relations between the latent variable models and several low-rank promoting penalty functions. The relations justify the use of Kronecker structured covariance matrices in a Gaussian based prior. In the methods, we use evidence approximation and expectation-maximization to learn the model parameters. The performance of the methods is evaluated through extensive numerical simulations.

We present the collaborative Kalman filter (CKF), a dynamic model for collaborative filtering and related factorization models. Using the matrix factorization approach to collaborative filtering, the CKF accounts for time evolution by modeling each low-dimensional latent embedding as a multidimensional Brownian motion. Each observation is a random variable whose distribution is parameterized by the dot product of the relevant Brownian motions at that moment in time. This is naturally interpreted as a Kalman filter with multiple interacting state space vectors. We also present a method for learning a dynamically evolving drift parameter for each location by modeling it as a geometric Brownian motion. We handle posterior intractability via a mean-field variational approximation, which also preserves tractability for downstream calculations in a manner similar to the Kalman filter. We evaluate the model on several large datasets, providing quantitative evaluation on the 10 million Movielens and 100 million Netflix datasets and qualitative evaluation on a set of 39 million stock returns divided across roughly 6,500 companies from the years 1962-2014.

Web services allow communication between heterogeneous systems in a distributed environment. Their enormous success and their increased use led to the fact that thousands of Web services are present on the Internet. This significant number of Web services which not cease to increase has led to problems of the difficulty in locating and classifying web services, these problems are encountered mainly during the operations of web services discovery and substitution . T raditional ways of search based on keywords are not successful in this context, their results do not support the structure of Web services and they consider in their search only the identifiers of the web service description language ( WSDL) interface elements. The methods based on semantics (WSDLS, OWLS, SAWSDL…) which increase the WSDL description of a Web service with a semantic description allow raising partially this problem, but their complexity and difficulty delays their adoption in real case s . Measuring the similarity between the web services interfaces is the most suitable solution for this kind of problems, it will classify available web services so as to know those that best match the searched profile and those that do not match. Thus, the main goal of this work is to study the degree of similarity between any two web services by offering a new method that is more effective than existing work s .

Sum Product Networks (SPNs) are a recently developed class of deep generative models which compute their associated unnormalized density functions using a special type of arithmetic circuit. When certain sufficient conditions, called the decomposability and completeness conditions (or "D&C" conditions), are imposed on the structure of these circuits, marginal densities and other useful quantities, which are typically intractable for other deep generative models, can be computed by what amounts to a single evaluation of the network (which is a property known as "validity"). However, the effect that the D&C conditions have on the capabilities of D&C SPNs is not well understood. In this work we analyze the D&C conditions, expose the various connections that D&C SPNs have with multilinear arithmetic circuits, and consider the question of how well they can capture various distributions as a function of their size and depth. Among our various contributions is a result which establishes the existence of a relatively simple distribution with fully tractable marginal densities which cannot be efficiently captured by D&C SPNs of any depth, but which can be efficiently captured by various other deep generative models. We also show that with each additional layer of depth permitted, the set of distributions which can be efficiently captured by D&C SPNs grows in size. This kind of "depth hierarchy" property has been widely conjectured to hold for various deep models, but has never been proven for any of them. Some of our other contributions include a new characterization of the D&C conditions as sufficient and necessary ones for a slightly strengthened notion of validity, and various state-machine characterizations of the types of computations that can be performed efficiently by D&C SPNs.

Hierarchical Bayesian networks and neural networks with stochastic hidden units are commonly perceived as two separate types of models. We show that either of these types of models can often be transformed into an instance of the other, by switching between centered and differentiable non-centered parameterizations of the latent variables. The choice of parameterization greatly influences the efficiency of gradient-based posterior inference; we show that they are often complementary to eachother, we clarify when each parameterization is preferred and show how inference can be made robust. In the non-centered form, a simple Monte Carlo estimator of the marginal likelihood can be used for learning the parameters. Theoretical results are supported by experiments.

This paper considers the problem of subspace clustering under noise. Specifically, we study the behavior of Sparse Subspace Clustering (SSC) when either adversarial or random noise is added to the unlabelled input data points, which are assumed to be in a union of low-dimensional subspaces. We show that a modified version of SSC is \emph{provably effective} in correctly identifying the underlying subspaces, even with noisy data. This extends theoretical guarantee of this algorithm to more practical settings and provides justification to the success of SSC in a class of real applications.

In response to the need for learning tools tuned to big data analytics, the present paper introduces a framework for efficient clustering of huge sets of (possibly high-dimensional) data. Building on random sampling and consensus (RANSAC) ideas pursued earlier in a different (computer vision) context for robust regression, a suite of novel dimensionality and set-reduction algorithms is developed. The advocated sketch-and-validate (SkeVa) family includes two algorithms that rely on K-means clustering per iteration on reduced number of dimensions and/or feature vectors: The first operates in a batch fashion, while the second sequential one offers computational efficiency and suitability with streaming modes of operation. For clustering even nonlinearly separable vectors, the SkeVa family offers also a member based on user-selected kernel functions. Further trading off performance for reduced complexity, a fourth member of the SkeVa family is based on a divergence criterion for selecting proper minimal subsets of feature variables and vectors, thus bypassing the need for K-means clustering per iteration. Extensive numerical tests on synthetic and real data sets highlight the potential of the proposed algorithms, and demonstrate their competitive performance relative to state-of-the-art random projection alternatives.

We consider the problem of uncertainty assessment for low dimensional components in high dimensional models. Specifically, we propose a decorrelated score function to handle the impact of high dimensional nuisance parameters. We consider both hypothesis tests and confidence regions for generic penalized M-estimators. Unlike most existing inferential methods which are tailored for individual models, our approach provides a general framework for high dimensional inference and is applicable to a wide range of applications. From the testing perspective, we develop general theorems to characterize the limiting distributions of the decorrelated score test statistic under both null hypothesis and local alternatives. These results provide asymptotic guarantees on the type I errors and local powers of the proposed test. Furthermore, we show that the decorrelated score function can be used to construct point and confidence region estimators that are semiparametrically efficient. We also generalize this framework to broaden its applications. First, we extend it to handle high dimensional null hypothesis, where the number of parameters of interest can increase exponentially fast with the sample size. Second, we establish the theory for model misspecification. Third, we go beyond the likelihood framework, by introducing the generalized score test based on general loss functions. Thorough numerical studies are conducted to back up the developed theoretical results.

High temporal resolution measurements of human brain activity can be performed by recording the electric potentials on the scalp surface (electroencephalography, EEG), or by recording the magnetic fields near the surface of the head (magnetoencephalography, MEG). The analysis of the data is problematic due to the fact that multiple neural generators may be simultaneously active and the potentials and magnetic fields from these sources are superimposed on the detectors. It is highly desirable to un-mix the data into signals representing the behaviors of the original individual generators. This general problem is called blind source separation and several recent techniques utilizing maximum entropy, minimum mutual information, and maximum likelihood estimation have been applied. These techniques have had much success in separating signals such as natural sounds or speech, but appear to be ineffective when applied to EEG or MEG signals. Many of these techniques implicitly assume that the source distributions have a large kurtosis, whereas an analysis of EEG/MEG signals reveals that the distributions are multimodal. This suggests that more effective separation techniques could be designed for EEG and MEG signals.