Several convex formulation methods have been proposed previously for statistical estimation with structured sparsity as the prior. These methods often require a carefully tuned regularization parameter, often a cumbersome or heuristic exercise. Furthermore, the estimate that these methods produce might not belong to the desired sparsity model, albeit accurately approximating the true parameter. Therefore, greedy-type algorithms could often be more desirable in estimating structured-sparse parameters. So far, these greedy methods have mostly focused on linear statistical models. In this paper we study the projected gradient descent with non-convex structured-sparse parameter model as the constraint set. Should the cost function have a Stable Model-Restricted Hessian the algorithm produces an approximation for the desired minimizer. As an example we elaborate on application of the main results to estimation in Generalized Linear Model.

Vector autoregression (VAR) is a fundamental tool for modeling the joint dynamics of multivariate time series. However, as the number of component series is increased, the VAR model quickly becomes overparameterized, making reliable estimation difficult and impeding its adoption as a forecasting tool in high dimensional settings. A number of authors have sought to address this issue by incorporating regularized approaches, such as the lasso, that impose sparse or low-rank structures on the estimated coefficient parameters of the VAR. More traditional approaches attempt to address overparameterization by selecting a low lag order, based on the assumption that dynamic dependence among components is short-range. However, these methods typically assume a single, universal lag order that applies across all components, unnecessarily constraining the dynamic relationship between the components and impeding forecast performance. The lasso-based approaches are more flexible but do not incorporate the notion of lag order selection. We propose a new class of regularized VAR models, called hierarchical vector autoregression (HVAR), that embed the notion of lag selection into a convex regularizer. The key convex modeling tool is a group lasso with nested groups which ensure the sparsity pattern of autoregressive lag coefficients honors the ordered structure inherent to VAR. We provide computationally efficient algorithms for solving HVAR problems that can be parallelized across the components. A simulation study shows the improved performance in forecasting and lag order selection over previous approaches, and a macroeconomic application further highlights forecasting improvements as well as the convenient, interpretable output of a HVAR model.

Identifying the type of font (e.g., Roman, Blackletter) used in historical documents can help optical character recognition (OCR) systems produce more accurate text transcriptions. Towards this end, we present an active-learning strategy that can significantly reduce the number of labeled samples needed to train a font classifier. Our approach extracts image-based features that exploit geometric differences between fonts at the word level, and combines them into a bag-of-word representation for each page in a document. We evaluate six sampling strategies based on uncertainty, dissimilarity and diversity criteria, and test them on a database containing over 3,000 historical documents with Blackletter, Roman and Mixed fonts. Our results show that a combination of uncertainty and diversity achieves the highest predictive accuracy (89% of test cases correctly classified) while requiring only a small fraction of the data (17%) to be labeled. We discuss the implications of this result for mass digitization projects of historical documents.

Scoring systems are linear classification models that only require users to add, subtract and multiply a few small numbers in order to make a prediction. These models are in widespread use by the medical community, but are difficult to learn from data because they need to be accurate and sparse, have coprime integer coefficients, and satisfy multiple operational constraints. We present a new method for creating data-driven scoring systems called a Supersparse Linear Integer Model (SLIM). SLIM scoring systems are built by solving an integer program that directly encodes measures of accuracy (the 0-1 loss) and sparsity (the $\ell_0$-seminorm) while restricting coefficients to coprime integers. SLIM can seamlessly incorporate a wide range of operational constraints related to accuracy and sparsity, and can produce highly tailored models without parameter tuning. We provide bounds on the testing and training accuracy of SLIM scoring systems, and present a new data reduction technique that can improve scalability by eliminating a portion of the training data beforehand. Our paper includes results from a collaboration with the Massachusetts General Hospital Sleep Laboratory, where SLIM was used to create a highly tailored scoring system for sleep apnea screening

In this paper we derive variability measures for the conditional probability distributions of a pair of random variables, and we study its application in the inference of causal-effect relationships. We also study the combination of the proposed measures with standard statistical measures in the the framework of the ChaLearn cause-effect pair challenge. The developed model obtains an AUC score of 0.82 on the final test database and ranked second in the challenge.

We study the problem of recovering a hidden community of cardinality $K$ from an $n \times n$ symmetric data matrix $A$, where for distinct indices $i,j$, $A_{ij} \sim P$ if $i, j$ both belong to the community and $A_{ij} \sim Q$ otherwise, for two known probability distributions $P$ and $Q$ depending on $n$. If $P={\rm Bern}(p)$ and $Q={\rm Bern}(q)$ with $p>q$, it reduces to the problem of finding a densely-connected $K$-subgraph planted in a large Erd\"os-R\'enyi graph; if $P=\mathcal{N}(\mu,1)$ and $Q=\mathcal{N}(0,1)$ with $\mu>0$, it corresponds to the problem of locating a $K \times K$ principal submatrix of elevated means in a large Gaussian random matrix. We focus on two types of asymptotic recovery guarantees as $n \to \infty$: (1) weak recovery: expected number of classification errors is $o(K)$; (2) exact recovery: probability of classifying all indices correctly converges to one. Under mild assumptions on $P$ and $Q$, and allowing the community size to scale sublinearly with $n$, we derive a set of sufficient conditions and a set of necessary conditions for recovery, which are asymptotically tight with sharp constants. The results hold in particular for the Gaussian case, and for the case of bounded log likelihood ratio, including the Bernoulli case whenever $\frac{p}{q}$ and $\frac{1-p}{1-q}$ are bounded away from zero and infinity. An important algorithmic implication is that, whenever exact recovery is information theoretically possible, any algorithm that provides weak recovery when the community size is concentrated near $K$ can be upgraded to achieve exact recovery in linear additional time by a simple voting procedure.

We study optimization algorithms based on variance reduction for stochastic gradient descent (SGD). Remarkable recent progress has been made in this direction through development of algorithms like SAG, SVRG, SAGA. These algorithms have been shown to outperform SGD, both theoretically and empirically. However, asynchronous versions of these algorithms---a crucial requirement for modern large-scale applications---have not been studied. We bridge this gap by presenting a unifying framework for many variance reduction techniques. Subsequently, we propose an asynchronous algorithm grounded in our framework, and prove its fast convergence. An important consequence of our general approach is that it yields asynchronous versions of variance reduction algorithms such as SVRG and SAGA as a byproduct. Our method achieves near linear speedup in sparse settings common to machine learning. We demonstrate the empirical performance of our method through a concrete realization of asynchronous SVRG.

Linear independence testing is a fundamental information-theoretic and statistical problem that can be posed as follows: given $n$ points $\{(X_i,Y_i)\}^n_{i=1}$ from a $p+q$ dimensional multivariate distribution where $X_i \in \mathbb{R}^p$ and $Y_i \in\mathbb{R}^q$, determine whether $a^T X$ and $b^T Y$ are uncorrelated for every $a \in \mathbb{R}^p, b\in \mathbb{R}^q$ or not. We give minimax lower bound for this problem (when $p+q,n \to \infty$, $(p+q)/n \leq \kappa < \infty$, without sparsity assumptions). In summary, our results imply that $n$ must be at least as large as $\sqrt {pq}/\|\Sigma_{XY}\|_F^2$ for any procedure (test) to have non-trivial power, where $\Sigma_{XY}$ is the cross-covariance matrix of $X,Y$. We also provide some evidence that the lower bound is tight, by connections to two-sample testing and regression in specific settings.

We propose a non-parametric anomaly detection algorithm for high dimensional data. We first rank scores derived from nearest neighbor graphs on $n$-point nominal training data. We then train limited complexity models to imitate these scores based on the max-margin learning-to-rank framework. A test-point is declared as an anomaly at $\alpha$-false alarm level if the predicted score is in the $\alpha$-percentile. The resulting anomaly detector is shown to be asymptotically optimal in that for any false alarm rate $\alpha$, its decision region converges to the $\alpha$-percentile minimum volume level set of the unknown underlying density. In addition, we test both the statistical performance and computational efficiency of our algorithm on a number of synthetic and real-data experiments. Our results demonstrate the superiority of our algorithm over existing $K$-NN based anomaly detection algorithms, with significant computational savings.

We present a system for online monitoring of maritime activity over streaming positions from numerous vessels sailing at sea. It employs an online tracking module for detecting important changes in the evolving trajectory of each vessel across time, and thus can incrementally retain concise, yet reliable summaries of its recent movement. In addition, thanks to its complex event recognition module, this system can also offer instant notification to marine authorities regarding emergency situations, such as risk of collisions, suspicious moves in protected zones, or package picking at open sea. Not only did our extensive tests validate the performance, efficiency, and robustness of the system against scalable volumes of real-world and synthetically enlarged datasets, but its deployment against online feeds from vessels has also confirmed its capabilities for effective, real-time maritime surveillance.