Recent advances in sensor technology have made it feasible to deploy a network of inexpensive sensors for carrying out synergistically even sophisticated inference tasks. In applications such as environmental monitoring, surveillance of critical infrastructure, agriculture, or medical imaging, the typical concept of operation involves a large and possibly heterogeneous set of sensors locally observing the signal of interest, and transmitting their measurements to a higher-layer agent (fusion center). This so-termed layered sensing apparatus entails three operational conditions: (c1) Each node's measurement vector comprising either a collection of scalar observations across time, or a snapshot of different sensor readings, is typically assumed to be linearly related to the unknown variable(s). Such a linear model can arise when the sensing system is viewed as a linear filter with known impulse response. Even when the underlying model is nonlinear, the observations are approximately modeled as adhering to a (multivariate) linear regression; (c2) Either because readings are costly to sense and transmit, due to delay or stationarity constraints, or simply because dimensionality reduction is invoked to cope with the "curse of dimensionality," the linear model is oftentimes under-determined, i.e., the dimension of the unknown vector is larger than that of each sensor's vector observation; and (c3) Not all sensors are reliable because failures in the sensing devices, fades of the sensor-agent communication link, physical obstruction of the scene of interest, and (un)intentional interference, all can severely deteriorate the consistency and reliability of sensor data.

We consider a collection of prediction experiments, which are clustered in the sense that groups of experiments ex- hibit similar relationship between the predictor and response variables. The experiment clusters as well as the regres- sion relationships are unknown. The regression relation- ships define the experiment clusters, and in general, the predictor and response variables may not exhibit any clus- tering. We call this prediction problem clustered regres- sion with unknown clusters (CRUC) and in this paper we focus on linear regression. We study and compare several methods for CRUC, demonstrate their applicability to the Yahoo Learning-to-rank Challenge (YLRC) dataset, and in- vestigate an associated mathematical model. CRUC is at the crossroads of many prior works and we study several prediction algorithms with diverse origins: an adaptation of the expectation-maximization algorithm, an approach in- spired by K-means clustering, the singular value threshold- ing approach to matrix rank minimization under quadratic constraints, an adaptation of the Curds and Whey method in multiple regression, and a local regression (LoR) scheme reminiscent of neighborhood methods in collaborative filter- ing. Based on empirical evaluation on the YLRC dataset as well as simulated data, we identify the LoR method as a good practical choice: it yields best or near-best prediction performance at a reasonable computational load, and it is less sensitive to the choice of the algorithm parameter. We also provide some analysis of the LoR method for an asso- ciated mathematical model, which sheds light on optimal parameter choice and prediction performance.

In multivariate regression, a $K$-dimensional response vector is regressed upon a common set of $p$ covariates, with a matrix $B^*\in\mathbb{R}^{p\times K}$ of regression coefficients. We study the behavior of the multivariate group Lasso, in which block regularization based on the $\ell_1/\ell_2$ norm is used for support union recovery, or recovery of the set of $s$ rows for which $B^*$ is nonzero. Under high-dimensional scaling, we show that the multivariate group Lasso exhibits a threshold for the recovery of the exact row pattern with high probability over the random design and noise that is specified by the sample complexity parameter $\theta(n,p,s):=n/[2\psi(B^*)\log(p-s)]$. Here $n$ is the sample size, and $\psi(B^*)$ is a sparsity-overlap function measuring a combination of the sparsities and overlaps of the $K$-regression coefficient vectors that constitute the model. We prove that the multivariate group Lasso succeeds for problem sequences $(n,p,s)$ such that $\theta(n,p,s)$ exceeds a critical level $\theta_u$, and fails for sequences such that $\theta(n,p,s)$ lies below a critical level $\theta_{\ell}$. For the special case of the standard Gaussian ensemble, we show that $\theta_{\ell}=\theta_u$ so that the characterization is sharp. The sparsity-overlap function $\psi(B^*)$ reveals that, if the design is uncorrelated on the active rows, $\ell_1/\ell_2$ regularization for multivariate regression never harms performance relative to an ordinary Lasso approach and can yield substantial improvements in sample complexity (up to a factor of $K$) when the coefficient vectors are suitably orthogonal. For more general designs, it is possible for the ordinary Lasso to outperform the multivariate group Lasso. We complement our analysis with simulations that demonstrate the sharpness of our theoretical results, even for relatively small problems.

The ensemble approach for statistical modelling was first made popular by such algorithms as boosting (Freund and Schapire 1996; Friedman et al. 2000), bagging (Breiman 1996), random forest (Breiman 2001), and the gradient boosting machine (Friedman 2001). They are powerful algorithms for solving prediction problems. This article is concerned with using the ensemble approach for a different problem, variable selection. We shall use the terms "prediction ensemble" and "variableselection ensemble" to differentiate ensembles used for these different purposes.

Under what conditions is an edge present in a social network at time t likely to decay or persist by some future time t + Delta(t)? Previous research addressing this issue suggests that the network range of the people involved in the edge, the extent to which the edge is embedded in a surrounding structure, and the age of the edge all play a role in edge decay. This paper uses weighted data from a large-scale social network built from cell-phone calls in an 8-week period to determine the importance of edge weight for the decay/persistence process. In particular, we study the relative predictive power of directed weight, embeddedness, newness, and range (measured as outdegree) with respect to edge decay and assess the effectiveness with which a simple decision tree and logistic regression classifier can accurately predict whether an edge that was active in one time period continues to be so in a future time period. We find that directed edge weight, weighted reciprocity and time-dependent measures of edge longevity are highly predictive of whether we classify an edge as persistent or decayed, relative to the other types of factors at the dyad and neighborhood level.

Although there is a rich literature on methods for allowing the variance in a univariate regression model to vary with predictors, time and other factors, relatively little has been done in the multivariate case. Our focus is on developing a class of nonparametric covariance regression models, which allow an unknown p x p covariance matrix to change flexibly with predictors. The proposed modeling framework induces a prior on a collection of covariance matrices indexed by predictors through priors for predictor-dependent loadings matrices in a factor model. In particular, the predictor-dependent loadings are characterized as a sparse combination of a collection of unknown dictionary functions (e.g, Gaussian process random functions). The induced covariance is then a regularized quadratic function of these dictionary elements. Our proposed framework leads to a highly-flexible, but computationally tractable formulation with simple conjugate posterior updates that can readily handle missing data. Theoretical properties are discussed and the methods are illustrated through simulations studies and an application to the Google Flu Trends data.

A variable screening procedure via correlation learning was proposed Fan and Lv (2008) to reduce dimensionality in sparse ultra-high dimensional models. Even when the true model is linear, the marginal regression can be highly nonlinear. To address this issue, we further extend the correlation learning to marginal nonparametric learning. Our nonparametric independence screening is called NIS, a specific member of the sure independence screening. Several closely related variable screening procedures are proposed. Under the nonparametric additive models, it is shown that under some mild technical conditions, the proposed independence screening methods enjoy a sure screening property. The extent to which the dimensionality can be reduced by independence screening is also explicitly quantified. As a methodological extension, an iterative nonparametric independence screening (INIS) is also proposed to enhance the finite sample performance for fitting sparse additive models. The simulation results and a real data analysis demonstrate that the proposed procedure works well with moderate sample size and large dimension and performs better than competing methods.

When choosing a suitable technique for regression and classification with multivariate predictor variables, one is often faced with a tradeoff between interpretability and high predictive accuracy. To give a classical example, classification and regression trees are easy to understand and interpret. Tree ensembles like Random Forests provide usually more accurate predictions. Yet tree ensembles are also more difficult to analyze than single trees and are often criticized, perhaps unfairly, as `black box' predictors. Node harvest is trying to reconcile the two aims of interpretability and predictive accuracy by combining positive aspects of trees and tree ensembles. Results are very sparse and interpretable and predictive accuracy is extremely competitive, especially for low signal-to-noise data. The procedure is simple: an initial set of a few thousand nodes is generated randomly. If a new observation falls into just a single node, its prediction is the mean response of all training observation within this node, identical to a tree-like prediction. A new observation falls typically into several nodes and its prediction is then the weighted average of the mean responses across all these nodes. The only role of node harvest is to `pick' the right nodes from the initial large ensemble of nodes by choosing node weights, which amounts in the proposed algorithm to a quadratic programming problem with linear inequality constraints. The solution is sparse in the sense that only very few nodes are selected with a nonzero weight. This sparsity is not explicitly enforced. Maybe surprisingly, it is not necessary to select a tuning parameter for optimal predictive accuracy. Node harvest can handle mixed data and missing values and is shown to be simple to interpret and competitive in predictive accuracy on a variety of data sets.

We analyze the convergence behaviour of a recently proposed algorithm for regularized estimation called Dual Augmented Lagrangian (DAL). Our analysis is based on a new interpretation of DAL as a proximal minimization algorithm. We theoretically show under some conditions that DAL converges super-linearly in a non-asymptotic and global sense. Due to a special modelling of sparse estimation problems in the context of machine learning, the assumptions we make are milder and more natural than those made in conventional analysis of augmented Lagrangian algorithms. In addition, the new interpretation enables us to generalize DAL to wide varieties of sparse estimation problems. We experimentally confirm our analysis in a large scale $\ell_1$-regularized logistic regression problem and extensively compare the efficiency of DAL algorithm to previously proposed algorithms on both synthetic and benchmark datasets.

We prove rates of convergence in the statistical sense for kernel-based least squares regression using a conjugate gradient algorithm, where regularization against overfitting is obtained by early stopping. This method is directly related to Kernel Partial Least Squares, a regression method that combines supervised dimensionality reduction with least squares projection. The rates depend on two key quantities: first, on the regularity of the target regression function and second, on the effective dimensionality of the data mapped into the kernel space. Lower bounds on attainable rates depending on these two quantities were established in earlier literature, and we obtain upper bounds for the considered method that match these lower bounds (up to a log factor) if the true regression function belongs to the reproducing kernel Hilbert space. If the latter assumption is not fulfilled, we obtain similar convergence rates provided additional unlabeled data are available. The order of the learning rates in these two situations match state-of-the-art results that were recently obtained for the least squares support vector machine and for linear regularization operators.