In compressed sensing, in order to recover a sparse or nearly sparse vector from possibly noisy measurements, the most popular approach is $\ell_1$-norm minimization. Upper bounds for the $\ell_2$- norm of the error between the true and estimated vectors are given in [1] and reviewed in [2], while bounds for the $\ell_1$-norm are given in [3]. When the unknown vector is not conventionally sparse but is "group sparse" instead, a variety of alternatives to the $\ell_1$-norm have been proposed in the literature, including the group LASSO, sparse group LASSO, and group LASSO with tree structured overlapping groups. However, no error bounds are available for any of these modified objective functions. In the present paper, a unified approach is presented for deriving upper bounds on the error between the true vector and its approximation, based on the notion of decomposable and $\gamma$-decomposable norms. The bounds presented cover all of the norms mentioned above, and also provide a guideline for choosing norms in future to accommodate alternate forms of sparsity.

Tomaso Poggio Center for Brains, Minds and Machines Massachusetts Institute of Technology tp@ai.mit.edu Learning to predict multi-label outputs is challenging, but in many problems there is a natural metric on the outputs that can be used to improve predictions. In this paper we develop a loss function for multi-label learning, based on the Wasserstein distance. The Wasserstein distance provides a natural notion of dissimilarity for probability measures. Although optimizing with respect to the exact Wasserstein distance is costly, recent work has described a regularized approximation that is efficiently computed. We describe an efficient learning algorithm based on this regularization, as well as a novel extension of the Wasserstein distance from probability measures to unnormalized measures. We also describe a statistical learning bound for the loss. The Wasserstein loss can encourage smoothness of the predictions with respect to a chosen metric on the output space. We demonstrate this property on a real-data tag prediction problem, using the Yahoo Flickr Creative Commons dataset, outperforming a baseline that doesn't use the metric.

Probabilistic Graphical Models (PGM) are very useful in the fields of machine learning and data mining. The crucial limitation of those models,however, is the scalability. The Bayesian Network, which is one of the most common PGMs used in machine learning and data mining, demonstrates this limitation when the training data consists of random variables, each of them has a large set of possible values. In the big data era, one would expect new extensions to the existing PGMs to handle the massive amount of data produced these days by computers, sensors and other electronic devices. With hierarchical data - data that is arranged in a treelike structure with several levels - one would expect to see hundreds of thousands or millions of values distributed over even just a small number of levels. When modeling this kind of hierarchical data across large data sets, Bayesian Networks become infeasible for representing the probability distributions. In this paper we introduce an extension to Bayesian Networks to handle massive sets of hierarchical data in a reasonable amount of time and space. The proposed model achieves perfect precision of 1.0 and high recall of 0.93 when it is used as multi-label classifier for the annotation of mass spectrometry data. On another data set of 1.5 billion search logs provided by CareerBuilder.com the model was able to predict latent semantic relationships between search keywords with accuracy up to 0.80.

We introduce a novel approach to Compressed Path Databases, space efficient oracles used to very quickly identify the first edge on a shortest path. Our algorithm achieves query running times on the 100 nanosecond scale, being significantly faster than state-of-the-art first-move oracles from the literature. Space consumption is competitive, due to a compression approach that rearranges rows and columns in a first-move matrix and then performs run length encoding (RLE) on the contents of the matrix. One variant of our implemented system was, by a convincing margin, the fastest entry in the 2014 Grid-Based Path Planning Competition. We give a first tractability analysis for the compression scheme used by our algorithm. We study the complexity of computing a database of minimum size for general directed and undirected graphs. We find that in both cases the problem is NP-complete. We also show that, for graphs which can be decomposed along articulation points, the problem can be decomposed into independent parts, with a corresponding reduction in its level of difficulty. In particular, this leads to simple and tractable algorithms with linear running time which yield optimal compression results for trees.

We study a regularizer which is defined as a parameterized infimum of quadratics, and which we call the box-norm. We show that the k-support norm, a regularizer proposed by Argyriou et al. (2012) for sparse vector prediction problems, belongs to this family, and the box-norm can be generated as a perturbation of the former. We derive an improved algorithm to compute the proximity operator of the squared box-norm, and we provide a method to compute the norm. We extend the norms to matrices, introducing the spectral k-support norm and spectral box-norm. We note that the spectral box-norm is essentially equivalent to the cluster norm, a multitask learning regularizer introduced by Jacob et al. (2009a), and which in turn can be interpreted as a perturbation of the spectral k-support norm. Centering the norm is important for multitask learning and we also provide a method to use centered versions of the norms as regularizers. Numerical experiments indicate that the spectral k-support and box-norms and their centered variants provide state of the art performance in matrix completion and multitask learning problems respectively.

The Hidden Markov Model (HMM) is one of the mainstays of statistical modeling of discrete time series, with applications including speech recognition, computational biology, computer vision and econometrics. Estimating an HMM from its observation process is often addressed via the Baum-Welch algorithm, which is known to be susceptible to local optima. In this paper, we first give a general characterization of the basin of attraction associated with any global optimum of the population likelihood. By exploiting this characterization, we provide non-asymptotic finite sample guarantees on the Baum-Welch updates, guaranteeing geometric convergence to a small ball of radius on the order of the minimax rate around a global optimum. As a concrete example, we prove a linear rate of convergence for a hidden Markov mixture of two isotropic Gaussians given a suitable mean separation and an initialization within a ball of large radius around (one of) the true parameters. To our knowledge, these are the first rigorous local convergence guarantees to global optima for the Baum-Welch algorithm in a setting where the likelihood function is nonconvex. We complement our theoretical results with thorough numerical simulations studying the convergence of the Baum-Welch algorithm and illustrating the accuracy of our predictions.

This paper addresses the problem of scalable optimization for L1-regularized conditional Gaussian graphical models. Conditional Gaussian graphical models generalize the well-known Gaussian graphical models to conditional distributions to model the output network influenced by conditioning input variables. While highly scalable optimization methods exist for sparse Gaussian graphical model estimation, state-of-the-art methods for conditional Gaussian graphical models are not efficient enough and more importantly, fail due to memory constraints for very large problems. In this paper, we propose a new optimization procedure based on a Newton method that efficiently iterates over two sub-problems, leading to drastic improvement in computation time compared to the previous methods. We then extend our method to scale to large problems under memory constraints, using block coordinate descent to limit memory usage while achieving fast convergence. Using synthetic and genomic data, we show that our methods can solve one million dimensional problems to high accuracy in a little over a day on a single machine.

Complicated generative models often result in a situation where computing the likelihood of observed data is intractable, while simulating from the conditional density given a parameter value is relatively easy. Approximate Bayesian Computation (ABC) is a paradigm that enables simulation-based posterior inference in such cases by measuring the similarity between simulated and observed data in terms of a chosen set of summary statistics. However, there is no general rule to construct sufficient summary statistics for complex models. Insufficient summary statistics will "leak" information, which leads to ABC algorithms yielding samples from an incorrect (partial) posterior. In this paper, we propose a fully nonparametric ABC paradigm which circumvents the need for manually selecting summary statistics. Our approach, K2-ABC, uses maximum mean discrepancy (MMD) to construct a dissimilarity measure between the observed and simulated data. The embedding of an empirical distribution of the data into a reproducing kernel Hilbert space plays a role of the summary statistic and is sufficient whenever the corresponding kernels are characteristic. Experiments on a simulated scenario and a real-world biological problem illustrate the effectiveness of the proposed algorithm. M Park and W Jitkrittum contributed equally.

Vehicles are becoming more and more connected, this opens up a larger attack surface which not only affects the passengers inside vehicles, but also people around them. These vulnerabilities exist because modern systems are built on the comparatively less secure and old CAN bus framework which lacks even basic authentication. Since a new protocol can only help future vehicles and not older vehicles, our approach tries to solve the issue as a data analytics problem and use machine learning techniques to secure cars. We develop a Hidden Markov Model to detect anomalous states from real data collected from vehicles. Using this model, while a vehicle is in operation, we are able to detect and issue alerts. Our model could be integrated as a plug-n-play device in all new and old cars.

The discovery of causal relationships from purely observational data is a fundamental problem in science. The most elementary form of such a causal discovery problem is to decide whether X causes Y or, alternatively, Y causes X, given joint observations of two variables X, Y. An example is to decide whether altitude causes temperature, or vice versa, given only joint measurements of both variables. Even under the simplifying assumptions of no confounding, no feedback loops, and no selection bias, such bivariate causal discovery problems are challenging. Nevertheless, several approaches for addressing those problems have been proposed in recent years. We review two families of such methods: Additive Noise Methods (ANM) and Information Geometric Causal Inference (IGCI). We present the benchmark CauseEffectPairs that consists of data for 100 different cause-effect pairs selected from 37 datasets from various domains (e.g., meteorology, biology, medicine, engineering, economy, etc.) and motivate our decisions regarding the "ground truth" causal directions of all pairs. We evaluate the performance of several bivariate causal discovery methods on these real-world benchmark data and in addition on artificially simulated data. Our empirical results on real-world data indicate that certain methods are indeed able to distinguish cause from effect using only purely observational data, although more benchmark data would be needed to obtain statistically significant conclusions. One of the best performing methods overall is the additive-noise method originally proposed by Hoyer et al. (2009), which obtains an accuracy of 63+-10 % and an AUC of 0.74+-0.05 on the real-world benchmark. As the main theoretical contribution of this work we prove the consistency of that method.