The CUR decomposition provides an approximation of a matrix X that has low reconstruction error and that is sparse in the sense that the resulting approximation lies in the span of only a few columns of X. In this regard, it appears to be similar to many sparse PCA methods. However, CUR takes a randomized algorithmic approach whereas most sparse PCA methods are framed as convex optimization problems. In this paper, we try to understand CUR from a sparse optimization viewpoint. In particular, we show that CUR is implicitly optimizing a sparse regression objective and, furthermore, cannot be directly cast as a sparse PCA method. We observe that the sparsity attained by CUR possesses an interesting structure, which leads us to formulate a sparse PCA method that achieves a CUR-like sparsity.

Heavy-tailed distributions naturally occur in many real life problems. Unfortunately, it is typically not possible to compute inference in closed-form in graphical models which involve such heavy tailed distributions. In this work, we propose a novel simple linear graphical model for independent latent random variables, called linear characteristic model (LCM), defined in the characteristic function domain. Using stable distributions, a heavy-tailed family of distributions which is a generalization of Cauchy, L\'evy and Gaussian distributions, we show for the first time, how to compute both exact and approximate inference in such a linear multivariate graphical model. LCMs are not limited to only stable distributions, in fact LCMs are always defined for any random variables (discrete, continuous or a mixture of both). We provide a realistic problem from the field of computer networks to demonstrate the applicability of our construction. Other potential application is iterative decoding of linear channels with non-Gaussian noise.

We present and analyze an agnostic active learning algorithm that works without keeping a version space. This is unlike all previous approaches where a restricted set of candidate hypotheses is maintained throughout learning, and only hypotheses from this set are ever returned. By avoiding this version space approach, our algorithm sheds the computational burden and brittleness associated with maintaining version spaces, yet still allows for substantial improvements over supervised learning for classification.

We consider the online binary classification problem, where we are given m classifiers. At each stage, the classifiers map the input to the probability that the input belongs to the positive class. An online classification meta-algorithm is an algorithm that combines the outputs of the classifiers in order to attain a certain goal, without having prior knowledge on the form and statistics of the input, and without prior knowledge on the performance of the given classifiers. In this paper, we use sensitivity and specificity as the performance metrics of the meta-algorithm. In particular, our goal is to design an algorithm which satisfies the following two properties (asymptotically): (i) its average false positive rate (fp-rate) is under some given threshold, and (ii) its average true positive rate (tp-rate) is not worse than the tp-rate of the best convex combination of the m given classifiers that satisfies fp-rate constraint, in hindsight. We show that this problem is in fact a special case of the regret minimization problem with constraints, and therefore the above goal is not attainable. Hence, we pose a relaxed goal and propose a corresponding practical online learning meta-algorithm that attains it. In the case of two classifiers, we show that this algorithm takes a very simple form. To our best knowledge, this is the first algorithm that addresses the problem of the average tp-rate maximization under average fp-rate constraints in the online setting.

Most current image categorization methods require large collections of manually annotated training examples to learn accurate visual recognition models. The time-consuming human labeling effort effectively limits these approaches to recognition problems involving a small number of different object classes. In order to address this shortcoming, in recent years several authors have proposed to learn object classifiers from weakly-labeled Internet images, such as photos retrieved by keyword-based image search engines. While this strategy eliminates the need for human supervision, the recognition accuracies of these methods are considerably lower than those obtained with fully-supervised approaches, because of the noisy nature of the labels associated to Web data. In this paper we investigate and compare methods that learn image classifiers by combining very few manually annotated examples (e.g., 1-10 images per class) and a large number of weakly-labeled Web photos retrieved using keyword-based image search. We cast this as a domain adaptation problem: given a few strongly-labeled examples in a target domain (the manually annotated examples) and many source domain examples (the weakly-labeled Web photos), learn classifiers yielding small generalization error on the target domain. Our experiments demonstrate that, for the same number of strongly-labeled examples, our domain adaptation approach produces significant recognition rate improvements over the best published results (e.g., 65% better when using 5 labeled training examples per class) and that our classifiers are one order of magnitude faster to learn and to evaluate than the best competing method, despite our use of large weakly-labeled data sets.

Accurate short-term wind forecasts (STWFs), with time horizons from 0.5 to 6 hours, are essential for efficient integration of wind power to the electrical power grid. Physical models based on numerical weather predictions are currently not competitive, and research on machine learning approaches is ongoing. Two major challenges confronting these efforts are missing observations and weather-regime induced dependency shifts among wind variables at geographically distributed sites. In this paper we introduce approaches that address both of these challenges. We describe a new regime-aware approach to STWF that use auto-regressive hidden Markov models (AR-HMM), a subclass of conditional linear Gaussian (CLG) models. Although AR-HMMs are a natural representation for weather regimes, as with CLG models in general, exact inference is NP-hard when observations are missing (Lerner and Parr, 2001). Because of this high cost, we introduce a simple approximate inference method for AR-HMMs, which we believe has applications to other sequential and temporal problem domains that involve continuous variables. In an empirical evaluation on publicly available wind data from two geographically distinct regions, our approach makes significantly more accurate predictions than baseline models, and uncovers meteorologically relevant regimes.

To cope with concept drift, we placed a probability distribution over the location of the most-recent drift point. We used Bayesian model comparison to update this distribution from the predictions of models trained on blocks of consecutive observations and pruned potential drift points with low probability. We compare our approach to a non-probabilistic method for drift and a probabilistic method for change-point detection. In our experiments, our approach generally yielded improved accuracy and/or speed over these other methods.

Sparse methods for supervised learning aim at finding good linear predictors from as few variables as possible, i.e., with small cardinality of their supports. This combinatorial selection problem is often turned into a convex optimization problem by replacing the cardinality function by its convex envelope (tightest convex lower bound), in this case the L1-norm. In this paper, we investigate more general set-functions than the cardinality, that may incorporate prior knowledge or structural constraints which are common in many applications: namely, we show that for nondecreasing submodular set-functions, the corresponding convex envelope can be obtained from its Lovasz extension, a common tool in submodular analysis. This defines a family of polyhedral norms, for which we provide generic algorithmic tools (subgradients and proximal operators) and theoretical results (conditions for support recovery or high-dimensional inference). By selecting specific submodular functions, we can give a new interpretation to known norms, such as those based on rank-statistics or grouped norms with potentially overlapping groups; we also define new norms, in particular ones that can be used as non-factorial priors for supervised learning.

Bayesian optimization methods are often used to optimize unknown functions that are costly to evaluate. Typically, these methods sequentially select inputs to be evaluated one at a time based on a posterior over the unknown function that is updated after each evaluation. There are a number of effective sequential policies for selecting the individual inputs. In many applications, however, it is desirable to perform multiple evaluations in parallel, which requires selecting batches of multiple inputs to evaluate at once. In this paper, we propose a novel approach to batch Bayesian optimization, providing a policy for selecting batches of inputs with the goal of optimizing the function as efficiently as possible. The key idea is to exploit the availability of high-quality and efficient sequential policies, by using Monte-Carlo simulation to select input batches that closely match their expected behavior. To the best of our knowledge, this is the first batch selection policy for Bayesian optimization. Our experimental results on six benchmarks show that the proposed approach significantly outperforms two baselines and can lead to large advantages over a top sequential approach in terms of performance per unit time.

Despite the ubiquity of clustering as a tool in unsupervised learning, there is not yet a consensus on a formal theory, and the vast majority of work in this direction has focused on unsupervised clustering. We study a recently proposed framework for supervised clustering where there is access to a teacher. We give an improved generic algorithm to cluster any concept class in that model. Our algorithm is query-efficient in the sense that it involves only a small amount of interaction with the teacher. We also present and study two natural generalizations of the model. The model assumes that the teacher response to the algorithm is perfect. We eliminate this limitation by proposing a noisy model and give an algorithm for clustering the class of intervals in this noisy model. We also propose a dynamic model where the teacher sees a random subset of the points. Finally, for datasets satisfying a spectrum of weak to strong properties, we give query bounds, and show that a class of clustering functions containing Single-Linkage will find the target clustering under the strongest property.