We consider energy minimization for undirected graphical models, also known as MAP-inference problem for Markov random fields. Although combinatorial methods, which return a provably optimal integral solution of the problem, made a big progress in the past decade, they are still typically unable to cope with large-scale datasets. On the other hand, large scale datasets are typically defined on sparse graphs, and convex relaxation methods, such as linear programming relaxations often provide good approximations to integral solutions. We propose a novel method of combining combinatorial and convex programming techniques to obtain a global solution of the initial combinatorial problem. Based on the information obtained from the solution of the convex relaxation, our method confines application of the combinatorial solver to a small fraction of the initial graphical model, which allows to optimally solve big problems. We demonstrate the power of our approach on a computer vision energy minimization benchmark.

Typical blur from camera shake often deviates from the standard uniform convolutional assumption, in part because of problematic rotations which create greater blurring away from some unknown center point. Consequently, successful blind deconvolution for removing shake artifacts requires the estimation of a spatially-varying or non-uniform blur operator. Using ideas from Bayesian inference and convex analysis, this paper derives a non-uniform blind deblurring algorithm with several desirable, yet previously-unexplored attributes. The underlying objective function includes a spatially-adaptive penalty that couples the latent sharp image, non-uniform blur operator, and noise level together. This coupling allows the penalty to automatically adjust its shape based on the estimated degree of local blur and image structure such that regions with large blur or few prominent edges are discounted. Remaining regions with modest blur and revealing edges therefore dominate the overall estimation process without explicitly incorporating structure-selection heuristics. The algorithm can be implemented using an optimization strategy that is virtually parameter free and simpler than existing methods. Detailed theoretical analysis and empirical validation on real images serve to validate the proposed method.

Label propagation is one of the state-of-the-art methods for semi-supervised learning, which estimates labels by propagating label information through a graph. Label propagation assumes that data points (nodes) connected in a graph should have similar labels. Consequently, the label estimation heavily depends on edge weights in a graph which represent similarity of each node pair. We propose a method for a graph to capture the manifold structure of input features using edge weights parameterized by a similarity function. In this approach, edge weights represent both similarity and local reconstruction weight simultaneously, both being reasonable for label propagation. For further justification, we provide analytical considerations including an interpretation as a cross-validation of a propagation model in the feature space, and an error analysis based on a low dimensional manifold model. Experimental results demonstrated the effectiveness of our approach both in synthetic and real datasets.

Many machine learning problems can be interpreted as learning for matching two types of objects (e.g., images and captions, users and products, queries and documents). The matching level of two objects is usually measured as the inner product in a certain feature space, while the modeling effort focuses on mapping of objects from the original space to the feature space. This schema, although proven successful on a range of matching tasks, is insufficient for capturing the rich structure in the matching process of more complicated objects. In this paper, we propose a new deep architecture to more effectively model the complicated matching relations between two objects from heterogeneous domains. More specifically, we apply this model to matching tasks in natural language, e.g., finding sensible responses for a tweet, or relevant answers to a given question. This new architecture naturally combines the localness and hierarchy intrinsic to the natural language problems, and therefore greatly improves upon the state-of-the-art models.

We introduce a novel formulation of multi-task learning (MTL) called parametric task learning (PTL) that can systematically handle infinitely many tasks parameterized by a continuous parameter. Our key finding is that, for a certain class of PTL problems, the path of optimal task-wise solutions can be represented as piecewise-linear functions of the continuous task parameter. Based on this fact, we employ a parametric programming technique to obtain the common shared representation across all the continuously parameterized tasks efficiently. We show that our PTL formulation is useful in various scenarios such as learning under non-stationarity, cost-sensitive learning, and quantile regression, and demonstrate the usefulness of the proposed method experimentally in these scenarios.

We propose a new class of structured Schatten norms for tensors that includes two recently proposed norms (overlapped'' and "latent'') for convex-optimization-based tensor decomposition. Based on the properties of the structured Schatten norms, we mathematically analyze the performance of "latent'' approach for tensor decomposition, which was empirically found to perform better than the "overlapped'' approach in some settings. We show theoretically that this is indeed the case. In particular, when the unknown true tensor is low-rank in a specific mode, this approach performs as well as knowing the mode with the smallest rank. Along the way, we show a novel duality result for structures Schatten norms, which is also interesting in the general context of structured sparsity. We confirm through numerical simulations that our theory can precisely predict the scaling behaviour of the mean squared error. "

The efficiency of Brain-Computer Interfaces (BCI) largely depends upon a reliable extraction of informative features from the high-dimensional EEG signal. A crucial step in this protocol is the computation of spatial filters. The Common Spatial Patterns (CSP) algorithm computes filters that maximize the difference in band power between two conditions, thus it is tailored to extract the relevant information in motor imagery experiments. However, CSP is highly sensitive to artifacts in the EEG data, i.e. few outliers may alter the estimate drastically and decrease classification performance. Inspired by concepts from the field of information geometry we propose a novel approach for robustifying CSP. More precisely, we formulate CSP as a divergence maximization problem and utilize the property of a particular type of divergence, namely beta divergence, for robustifying the estimation of spatial filters in the presence of artifacts in the data. We demonstrate the usefulness of our method on toy data and on EEG recordings from 80 subjects.

In this paper, we propose a new and computationally efficient framework for learning sparse models. We formulate a unified approach that contains as particular cases models promoting sparse synthesis and analysis type of priors, and mixtures thereof. The supervised training of the proposed model is formulated as a bilevel optimization problem, in which the operators are optimized to achieve the best possible performance on a specific task, e.g., reconstruction or classification. By restricting the operators to be shift invariant, our approach can be thought as a way of learning analysis+synthesis sparsity-promoting convolutional operators. Leveraging recent ideas on fast trainable regressors designed to approximate exact sparse codes, we propose a way of constructing feed-forward neural networks capable of approximating the learned models at a fraction of the computational cost of exact solvers. In the shift-invariant case, this leads to a principled way of constructing task-specific convolutional networks. We illustrate the proposed models on several experiments in music analysis and image processing applications.

We study exploration in Multi-Armed Bandits (MAB) in a setting where~$k$ players collaborate in order to identify an $\epsilon$-optimal arm. Our motivation comes from recent employment of MAB algorithms in computationally intensive, large-scale applications. Our results demonstrate a non-trivial tradeoff between the number of arm pulls required by each of the players, and the amount of communication between them. In particular, our main result shows that by allowing the $k$ players to communicate \emph{only once}, they are able to learn $\sqrt{k}$ times faster than a single player. That is, distributing learning to $k$ players gives rise to a factor~$\sqrt{k}$ parallel speed-up. We complement this result with a lower bound showing this is in general the best possible. On the other extreme, we present an algorithm that achieves the ideal factor $k$ speed-up in learning performance, with communication only logarithmic in~$1/\epsilon$.

We consider the online Principal Component Analysis (PCA) where contaminated samples (containing outliers) are revealed sequentially to the Principal Components (PCs)estimator. Due to their sensitiveness to outliers, previous online PCA algorithms fail in this case and their results can be arbitrarily skewed by the outliers. Herewe propose the online robust PCA algorithm, which is able to improve the PCs estimation upon an initial one steadily, even when faced with a constant fraction of outliers. We show that the final result of the proposed online RPCA has an acceptable degradation from the optimum. Actually, under mild conditions, online RPCA achieves the maximal robustness with a 50% breakdown point. Moreover, online RPCA is shown to be efficient for both storage and computation, sinceit need not re-explore the previous samples as in traditional robust PCA algorithms.