For massive data sets, efficient computation commonly relies on distributed algorithms that store and process subsets of the data on different machines, minimizing communication costs. Our focus is on regression and classification problems involving many features. A variety of distributed algorithms have been proposed in this context, but challenges arise in defining an algorithm with low communication, theoretical guarantees and excellent practical performance in general settings. We propose a MEdian Selection Subset AGgregation Estimator (message) algorithm, which attempts to solve these problems. The algorithm applies feature selection in parallel for each subset using Lasso or another method, calculates the `median' feature inclusion index, estimates coefficients for the selected features in parallel for each subset, and then averages these estimates. The algorithm is simple, involves very minimal communication, scales efficiently in both sample and feature size, and has theoretical guarantees. In particular, we show model selection consistency and coefficient estimation efficiency. Extensive experiments show excellent performance in variable selection, estimation, prediction, and computation time relative to usual competitors.

Restricted Boltzmann machines (RBMs) are powerful machine learning models, but learning and some kinds of inference in the model require sampling-based approximations, which, in classical digital computers, are implemented using expensive MCMC. Physical computation offers the opportunity to reduce the cost of sampling by building physical systems whose natural dynamics correspond to drawing samples from the desired RBM distribution. Such a system avoids the burn-in and mixing cost of a Markov chain. However, hardware implementations of this variety usually entail limitations such as low-precision and limited range of the parameters and restrictions on the size and topology of the RBM. We conduct software simulations to determine how harmful each of these restrictions is. Our simulations are designed to reproduce aspects of the D-Wave quantum computer, but the issues we investigate arise in most forms of physical computation.

In this paper, we present an infinite hierarchical non-parametric Bayesian model to extract the hidden factors over observed data, where the number of hidden factors for each layer is unknown and can be potentially infinite. Moreover, the number of layers can also be infinite. We construct the model structure that allows continuous values for the hidden factors and weights, which makes the model suitable for various applications. We use the Metropolis-Hastings method to infer the model structure. Then the performance of the algorithm is evaluated by the experiments. Simulation results show that the model fits the underlying structure of simulated data.

We are focusing on bound constrained global optimization problems, whose objective functions are computationally expensive black-box functions and have multiple local minima. The recently popular Metric Stochastic Response Surface (MSRS) algorithm proposed by \cite{Regis2007SRBF} based on adaptive or sequential learning based on response surfaces is revisited and further extended for better performance in case of higher dimensional problems. Specifically, we propose a new way to generate the candidate points which the next function evaluation point is picked from according to the metric criteria, based on a new definition of distance, and prove the global convergence of the corresponding. Correspondingly, a more adaptive implementation of MSRS, named "SO-SA", is presented. "SO-SA" is is more likely to perturb those most sensitive coordinates when generating the candidate points, instead of perturbing all coordinates simultaneously. Numerical experiments on both synthetic problems and real problems demonstrate the advantages of our new algorithm, compared with many state of the art alternatives.}

In this paper we analyze a budgeted learning setting, in which the learner can only choose and observe a small subset of the attributes of each training example. We develop efficient algorithms for ridge and lasso linear regression, which utilize the geometry of the data by a novel data-dependent sampling scheme. When the learner has prior knowledge on the second moments of the attributes, the optimal sampling probabilities can be calculated precisely, and result in data-dependent improvements factors for the excess risk over the state-of-the-art that may be as large as $O(\sqrt{d})$, where $d$ is the problem's dimension. Moreover, under reasonable assumptions our algorithms can use less attributes than full-information algorithms, which is the main concern in budgeted learning settings. To the best of our knowledge, these are the first algorithms able to do so in our setting. Where no such prior knowledge is available, we develop a simple estimation technique that given a sufficient amount of training examples, achieves similar improvements. We complement our theoretical analysis with experiments on several data sets which support our claims.

Low-rank modeling generally refers to a class of methods that solve problems by representing variables of interest as low-rank matrices. It has achieved great success in various fields including computer vision, data mining, signal processing and bioinformatics. Recently, much progress has been made in theories, algorithms and applications of low-rank modeling, such as exact low-rank matrix recovery via convex programming and matrix completion applied to collaborative filtering. These advances have brought more and more attentions to this topic. In this paper, we review the recent advance of low-rank modeling, the state-of-the-art algorithms, and related applications in image analysis. We first give an overview to the concept of low-rank modeling and challenging problems in this area. Then, we summarize the models and algorithms for low-rank matrix recovery and illustrate their advantages and limitations with numerical experiments. Next, we introduce a few applications of low-rank modeling in the context of image analysis. Finally, we conclude this paper with some discussions.

Low-rank matrix estimation from incomplete measurements recently received increased attention due to the emergence of several challenging applications, such as recommender systems; see in particular the famous Netflix challenge. While the behaviour of algorithms based on nuclear norm minimization is now well understood, an as yet unexplored avenue of research is the behaviour of Bayesian algorithms in this context. In this paper, we briefly review the priors used in the Bayesian literature for matrix completion. A standard approach is to assign an inverse gamma prior to the singular values of a certain singular value decomposition of the matrix of interest; this prior is conjugate. However, we show that two other types of priors (again for the singular values) may be conjugate for this model: a gamma prior, and a discrete prior. Conjugacy is very convenient, as it makes it possible to implement either Gibbs sampling or Variational Bayes. Interestingly enough, the maximum a posteriori for these different priors is related to the nuclear norm minimization problems. We also compare all these priors on simulated datasets, and on the classical MovieLens and Netflix datasets.

Neurons in higher cortical areas, such as the prefrontal cortex, are known to be tuned to a variety of sensory and motor variables. The resulting diversity of neural tuning often obscures the represented information. Here we introduce a novel dimensionality reduction technique, demixed principal component analysis (dPCA), which automatically discovers and highlights the essential features in complex population activities. We reanalyze population data from the prefrontal areas of rats and monkeys performing a variety of working memory and decision-making tasks. In each case, dPCA summarizes the relevant features of the population response in a single figure. The population activity is decomposed into a few demixed components that capture most of the variance in the data and that highlight dynamic tuning of the population to various task parameters, such as stimuli, decisions, rewards, etc. Moreover, dPCA reveals strong, condition-independent components of the population activity that remain unnoticed with conventional approaches.

We propose a new active learning algorithm for parametric linear regression with random design. We provide finite sample convergence guarantees for general distributions in the misspecified model. This is the first active learner for this setting that provably can improve over passive learning. Unlike other learning settings (such as classification), in regression the passive learning rate of $O(1/\epsilon)$ cannot in general be improved upon. Nonetheless, the so-called `constant' in the rate of convergence, which is characterized by a distribution-dependent risk, can be improved in many cases. For a given distribution, achieving the optimal risk requires prior knowledge of the distribution. Following the stratification technique advocated in Monte-Carlo function integration, our active learner approaches the optimal risk using piecewise constant approximations.

A wide variety of application domains are concerned with data consisting of entities and their relationships or connections, formally represented as graphs. Within these diverse application areas, a common problem of interest is the detection of a subset of entities whose connectivity is anomalous with respect to the rest of the data. While the detection of such anomalous subgraphs has received a substantial amount of attention, no application-agnostic framework exists for analysis of signal detectability in graph-based data. In this paper, we describe a framework that enables such analysis using the principal eigenspace of a graph's residuals matrix, commonly called the modularity matrix in community detection. Leveraging this analytical tool, we show that the framework has a natural power metric in the spectral norm of the anomalous subgraph's adjacency matrix (signal power) and of the background graph's residuals matrix (noise power). We propose several algorithms based on spectral properties of the residuals matrix, with more computationally expensive techniques providing greater detection power. Detection and identification performance are presented for a number of signal and noise models, including clusters and bipartite foregrounds embedded into simple random backgrounds as well as graphs with community structure and realistic degree distributions. The trends observed verify intuition gleaned from other signal processing areas, such as greater detection power when the signal is embedded within a less active portion of the background. We demonstrate the utility of the proposed techniques in detecting small, highly anomalous subgraphs in real graphs derived from Internet traffic and product co-purchases.