We consider the task of estimating a Gaussian graphical model in the high-dimensional setting. The graphical lasso, which involves maximizing the Gaussian log likelihood subject to an l1 penalty, is a well-studied approach for this task. We begin by introducing a surprising connection between the graphical lasso and hierarchical clustering: the graphical lasso in effect performs a two-step procedure, in which (1) single linkage hierarchical clustering is performed on the variables in order to identify connected components, and then (2) an l1-penalized log likelihood is maximized on the subset of variables within each connected component. In other words, the graphical lasso determines the connected components of the estimated network via single linkage clustering. Unfortunately, single linkage clustering is known to perform poorly in certain settings. Therefore, we propose the cluster graphical lasso, which involves clustering the features using an alternative to single linkage clustering, and then performing the graphical lasso on the subset of variables within each cluster. We establish model selection consistency for this technique, and demonstrate its improved performance relative to the graphical lasso in a simulation study, as well as in applications to an equities data set, a university webpage data set, and a gene expression data set.

We develop a new model and algorithms for machine learning-based learning analytics, which estimate a learner's knowledge of the concepts underlying a domain, and content analytics, which estimate the relationships among a collection of questions and those concepts. Our model represents the probability that a learner provides the correct response to a question in terms of three factors: their understanding of a set of underlying concepts, the concepts involved in each question, and each question's intrinsic difficulty. We estimate these factors given the graded responses to a collection of questions. The underlying estimation problem is ill-posed in general, especially when only a subset of the questions are answered. The key observation that enables a well-posed solution is the fact that typical educational domains of interest involve only a small number of key concepts. Leveraging this observation, we develop both a bi-convex maximum-likelihood and a Bayesian solution to the resulting SPARse Factor Analysis (SPARFA) problem. We also incorporate user-defined tags on questions to facilitate the interpretability of the estimated factors. Experiments with synthetic and real-world data demonstrate the efficacy of our approach. Finally, we make a connection between SPARFA and noisy, binary-valued (1-bit) dictionary learning that is of independent interest.

The l1-regularized logistic regression (or sparse logistic regression) is a widely used method for simultaneous classification and feature selection. Although many recent efforts have been devoted to its efficient implementation, its application to high dimensional data still poses significant challenges. In this paper, we present a fast and effective sparse logistic regression screening rule (Slores) to identify the 0 components in the solution vector, which may lead to a substantial reduction in the number of features to be entered to the optimization. An appealing feature of Slores is that the data set needs to be scanned only once to run the screening and its computational cost is negligible compared to that of solving the sparse logistic regression problem. Moreover, Slores is independent of solvers for sparse logistic regression, thus Slores can be integrated with any existing solver to improve the efficiency. We have evaluated Slores using high-dimensional data sets from different applications. Extensive experimental results demonstrate that Slores outperforms the existing state-of-the-art screening rules and the efficiency of solving sparse logistic regression is improved by one magnitude in general.

We consider the problem of clustering noisy high-dimensional data points into a union of low-dimensional subspaces and a set of outliers. The number of subspaces, their dimensions, and their orientations are unknown. A probabilistic performance analysis of the thresholding-based subspace clustering (TSC) algorithm introduced recently in [1] shows that TSC succeeds in the noisy case, even when the subspaces intersect. Our results reveal an explicit tradeoff between the allowed noise level and the affinity of the subspaces. We furthermore find that the simple outlier detection scheme introduced in [1] provably succeeds in the noisy case.

When considering the problem of unmixing hyperspectral images, most of the literature in the geoscience and image processing areas relies on the widely used linear mixing model (LMM). However, the LMM may be not valid and other nonlinear models need to be considered, for instance, when there are multi-scattering effects or intimate interactions. Consequently, over the last few years, several significant contributions have been proposed to overcome the limitations inherent in the LMM. In this paper, we present an overview of recent advances in nonlinear unmixing modeling.

The marginal maximum a posteriori probability (MAP) estimation problem, which calculates the mode of the marginal posterior distribution of a subset of variables with the remaining variables marginalized, is an important inference problem in many models, such as those with hidden variables or uncertain parameters. Unfortunately, marginal MAP can be NP-hard even on trees, and has attracted less attention in the literature compared to the joint MAP (maximization) and marginalization problems. We derive a general dual representation for marginal MAP that naturally integrates the marginalization and maximization operations into a joint variational optimization problem, making it possible to easily extend most or all variational-based algorithms to marginal MAP. In particular, we derive a set of "mixed-product" message passing algorithms for marginal MAP, whose form is a hybrid of max-product, sum-product and a novel "argmax-product" message updates. We also derive a class of convergent algorithms based on proximal point methods, including one that transforms the marginal MAP problem into a sequence of standard marginalization problems. Theoretically, we provide guarantees under which our algorithms give globally or locally optimal solutions, and provide novel upper bounds on the optimal objectives. Empirically, we demonstrate that our algorithms significantly outperform the existing approaches, including a state-of-the-art algorithm based on local search methods.

Sparse learning has recently received increasing attention in many areas including machine learning, statistics, and applied mathematics. The mixed-norm regularization based on the l1q norm with q>1 is attractive in many applications of regression and classification in that it facilitates group sparsity in the model. The resulting optimization problem is, however, challenging to solve due to the inherent structure of the mixed-norm regularization. Existing work deals with special cases with q=1, 2, infinity, and they cannot be easily extended to the general case. In this paper, we propose an efficient algorithm based on the accelerated gradient method for solving the general l1q-regularized problem. One key building block of the proposed algorithm is the l1q-regularized Euclidean projection (EP_1q). Our theoretical analysis reveals the key properties of EP_1q and illustrates why EP_1q for the general q is significantly more challenging to solve than the special cases. Based on our theoretical analysis, we develop an efficient algorithm for EP_1q by solving two zero finding problems. To further improve the efficiency of solving large dimensional mixed-norm regularized problems, we propose a screening method which is able to quickly identify the inactive groups, i.e., groups that have 0 components in the solution. This may lead to substantial reduction in the number of groups to be entered to the optimization. An appealing feature of our screening method is that the data set needs to be scanned only once to run the screening. Compared to that of solving the mixed-norm regularized problems, the computational cost of our screening test is negligible. The key of the proposed screening method is an accurate sensitivity analysis of the dual optimal solution when the regularization parameter varies. Experimental results demonstrate the efficiency of the proposed algorithm.

We introduce a conceptually novel structured prediction model, GPstruct, which is kernelized, non-parametric and Bayesian, by design. We motivate the model with respect to existing approaches, among others, conditional random fields (CRFs), maximum margin Markov networks (M3N), and structured support vector machines (SVMstruct), which embody only a subset of its properties. We present an inference procedure based on Markov Chain Monte Carlo. The framework can be instantiated for a wide range of structured objects such as linear chains, trees, grids, and other general graphs. As a proof of concept, the model is benchmarked on several natural language processing tasks and a video gesture segmentation task involving a linear chain structure. We show prediction accuracies for GPstruct which are comparable to or exceeding those of CRFs and SVMstruct.

We explore the effect of introducing prior information into the intermediate level of neural networks for a learning task on which all the state-of-the-art machine learning algorithms tested failed to learn. We motivate our work from the hypothesis that humans learn such intermediate concepts from other individuals via a form of supervision or guidance using a curriculum. The experiments we have conducted provide positive evidence in favor of this hypothesis. In our experiments, a two-tiered MLP architecture is trained on a dataset with 64x64 binary inputs images, each image with three sprites. The final task is to decide whether all the sprites are the same or one of them is different. Sprites are pentomino tetris shapes and they are placed in an image with different locations using scaling and rotation transformations. The first part of the two-tiered MLP is pre-trained with intermediate-level targets being the presence of sprites at each location, while the second part takes the output of the first part as input and predicts the final task's target binary event. The two-tiered MLP architecture, with a few tens of thousand examples, was able to learn the task perfectly, whereas all other algorithms (include unsupervised pre-training, but also traditional algorithms like SVMs, decision trees and boosting) all perform no better than chance. We hypothesize that the optimization difficulty involved when the intermediate pre-training is not performed is due to the {\em composition} of two highly non-linear tasks. Our findings are also consistent with hypotheses on cultural learning inspired by the observations of optimization problems with deep learning, presumably because of effective local minima.

Graph comparison plays a major role in many network applications. We often need a similarity metric for comparing networks according to their structural properties. Various network features - such as degree distribution and clustering coefficient - provide measurements for comparing networks from different points of view, but a global and integrated distance metric is still missing. In this paper, we employ distance metric learning algorithms in order to construct an integrated distance metric for comparing structural properties of complex networks. According to natural witnesses of network similarities (such as network categories) the distance metric is learned by the means of a dataset of some labeled real networks. For evaluating our proposed method which is called NetDistance, we applied it as the distance metric in K-nearest-neighbors classification. Empirical results show that NetDistance outperforms previous methods, at least 20 percent, with respect to precision.