We introduce a new family of matrix norms, the ''local max'' norms, generalizing existing methods such as the max norm, the trace norm (nuclear norm), and the weighted or smoothed weighted trace norms, which have been extensively used in the literature as regularizers for matrix reconstruction problems. We show that this new family can be used to interpolate between the (weighted or unweighted) trace norm and the more conservative max norm. We test this interpolation on simulated data and on the large-scale Netflix and MovieLens ratings data, and find improved accuracy relative to the existing matrix norms. We also provide theoretical results showing learning guarantees for some of the new norms.

Given $\alpha,\epsilon$, we study the time complexity required to improperly learn a halfspace with misclassification error rate of at most $(1+\alpha)\,L^*_\gamma + \epsilon$, where $L^*_\gamma$ is the optimal $\gamma$-margin error rate. For $\alpha = 1/\gamma$, polynomial time and sample complexity is achievable using the hinge-loss. For $\alpha = 0$, \cite{ShalevShSr11} showed that $\poly(1/\gamma)$ time is impossible, while learning is possible in time $\exp(\tilde{O}(1/\gamma))$. An immediate question, which this paper tackles, is what is achievable if $\alpha \in (0,1/\gamma)$. We derive positive results interpolating between the polynomial time for $\alpha = 1/\gamma$ and the exponential time for $\alpha=0$. In particular, we show that there are cases in which $\alpha = o(1/\gamma)$ but the problem is still solvable in polynomial time. Our results naturally extend to the adversarial online learning model and to the PAC learning with malicious noise model.

The task of assigning a set of relevant tags to an image is challenging due to the size and variability of tag vocabularies. Consequently, most existing algorithms focus on tag assignment and fix an often large number of hand-crafted features to describe image characteristics. In this paper we introduce a hierarchical model for learning representations of full sized color images from the pixel level, removing the need for engineered feature representations and subsequent feature selection. We benchmark our model on the STL-10 recognition dataset, achieving state-of-the-art performance. When our features are combined with TagProp (Guillaumin et al.), we outperform or compete with existing annotation approaches that use over a dozen distinct image descriptors. Furthermore, using 256-bit codes and Hamming distance for training TagProp, we exchange only a small reduction in performance for efficient storage and fast comparisons. In our experiments, using deeper architectures always outperform shallow ones.

The empirical success of the belief propagation approximate inference algorithm has inspired numerous theoretical and algorithmic advances. Yet, for continuous non-Gaussian domains performing belief propagation remains a challenging task: recent innovations such as nonparametric or kernel belief propagation, while useful, come with a substantial computational cost and offer little theoretical guarantees, even for tree structured models. In this work we present Nonparanormal BP for performing efficient inference on distributions parameterized by a Gaussian copulas network and any univariate marginals. For tree structured networks, our approach is guaranteed to be exact for this powerful class of non-Gaussian models. Importantly, the method is as efficient as standard Gaussian BP, and its convergence properties do not depend on the complexity of the univariate marginals, even when a nonparametric representation is used.

The ability to predict action content from neural signals in real time before the action occurshas been long sought in the neuroscientific study of decision-making, agency and volition. Online real-time (ORT) prediction is important for understanding therelation between neural correlates of decision-making and conscious, voluntary action as well as for brain-machine interfaces. Here, epilepsy patients, implanted with intracranial depth microelectrodes or subdural grid electrodes for clinical purposes, participated in a "matching-pennies" game against an opponent. In each trial, subjects were given a 5 s countdown, after which they had to raise their left or right hand immediately as the "go" signal appeared on a computer screen. They won a fixed amount of money if they raised a different hand than their opponent and lost that amount otherwise.

This paper concerns the problem of matrix completion, which is to estimate a matrix from observations in a small subset of indices. We propose a calibrated spectrum elastic net method with a sum of the nuclear and Frobenius penalties and develop an iterative algorithm to solve the convex minimization problem. The iterative algorithm alternates between imputing the missing entries in the incomplete matrix by the current guess and estimating the matrix by a scaled soft-thresholding singular value decomposition of the imputed matrix until the resulting matrix converges. A calibration step follows to correct the bias caused by the Frobenius penalty. Under proper coherence conditions and for suitable penalties levels, we prove that the proposed estimator achieves an error bound of nearly optimal order and in proportion to the noise level. This provides a unified analysis of the noisy and noiseless matrix completion problems. Simulation results are presented to compare our proposal with previous ones.

We consider the problem of identifying patterns in a data set that exhibit anomalous behavior, often referred to as anomaly detection. In most anomaly detection algorithms, the dissimilarity between data samples is calculated by a single criterion, such as Euclidean distance. However, in many cases there may not exist a single dissimilarity measure that captures all possible anomalous patterns. In such a case, multiple criteria can be defined, and one can test for anomalies by scalarizing the multiple criteria by taking some linear combination of them. If the importance of the different criteria are not known in advance, the algorithm may need to be executed multiple times with different choices of weights in the linear combination. In this paper, we introduce a novel non-parametric multi-criteria anomaly detection method using Pareto depth analysis (PDA). PDA uses the concept of Pareto optimality to detect anomalies under multiple criteria without having to run an algorithm multiple times with different choices of weights. The proposed PDA approach scales linearly in the number of criteria and is provably better than linear combinations of the criteria.

Latent SVMs (LSVMs) are a class of powerful tools that have been successfully applied to many applications in computer vision. However, a limitation of LSVMs is that they rely on linear models. For many computer vision tasks, linear models are suboptimal and nonlinear models learned with kernels typically perform much better. Therefore it is desirable to develop the kernel version of LSVM. In this paper, we propose kernel latent SVM (KLSVM) -- a new learning framework that combines latent SVMs and kernel methods. We develop an iterative training algorithm to learn the model parameters. We demonstrate the effectiveness of KLSVM using three different applications in visual recognition. Our KLSVM formulation is very general and can be applied to solve a wide range of applications in computer vision and machine learning.

We advocate the use of a new distribution family--the transelliptical--for robust inference of high dimensional graphical models. The transelliptical family is an extension of the nonparanormal family proposed by Liu et al. (2009). Just as the nonparanormal extends the normal by transforming the variables using univariate functions, the transelliptical extends the elliptical family in the same way. We propose a nonparametric rank-based regularization estimator which achieves the parametric rates of convergence for both graph recovery and parameter estimation. Sucha result suggests that the extra robustness and flexibility obtained by the semiparametric transelliptical modeling incurs almost no efficiency loss. We also discuss the relationship between this work with the transelliptical component analysis proposed by Han and Liu (2012).

Graphical models are a very useful tool to describe and understand natural phenomena, from gene expression to climate change and social interactions. The topological structure of these graphs/networks is a fundamental part of the analysis, and in many cases the main goal of the study. However, little work has been done on incorporating prior topological knowledge onto the estimation of the underlying graphical models from sample data. In this work we propose extensions to the basic joint regression model for network estimation, which explicitly incorporate graph-topological constraints into the corresponding optimization approach. The first proposed extension includes an eigenvector centrality constraint, thereby promoting this important prior topological property. The second developed extension promotes the formation of certain motifs, triangle-shaped ones in particular, which are known to exist for example in genetic regulatory networks. The presentation of the underlying formulations, which serve as examples of the introduction of topological constraints in network estimation, is complemented with examples in diverse datasets demonstrating the importance of incorporating such critical prior knowledge.