Sparse coding consists in representing signals as sparse linear combinations of atoms selected from a dictionary. We consider an extension of this framework where the atoms are further assumed to be embedded in a tree. This is achieved using a recently introduced tree-structured sparse regularization norm, which has proven useful in several applications. This norm leads to regularized problems that are difficult to optimize, and we propose in this paper efficient algorithms for solving them. More precisely, we show that the proximal operator associated with this norm is computable exactly via a dual approach that can be viewed as the composition of elementary proximal operators. Our procedure has a complexity linear, or close to linear, in the number of atoms, and allows the use of accelerated gradient techniques to solve the tree-structured sparse approximation problem at the same computational cost as traditional ones using the ℓ1-norm. Our method is efficient and scales gracefully to millions of variables, which we illustrate in two types of applications: first, we consider fixed hierarchical dictionaries of wavelets to denoise natural images. Then, we apply our optimization tools in the context of dictionary learning, where learned dictionary elements naturally organize in a prespecified arborescent structure, leading to a better performance in reconstruction of natural image patches. When applied to text documents, our method learns hierarchies of topics, thus providing a competitive alternative to probabilistic topic models.

We consider a class of learning problems regularized by a structured sparsity-inducing norm defined as the sum of ℓ2- or ℓ∞-norms over groups of variables. Whereas much effort has been put in developing fast optimization techniques when the groups are disjoint or embedded in a hierarchy, we address here the case of general overlapping groups. To this end, we present two different strategies: On the one hand, we show that the proximal operator associated with a sum of ℓ∞norms can be computed exactly in polynomial time by solving a quadratic min-cost flow problem, allowing the use of accelerated proximal gradient methods. On the other hand, we use proximal splitting techniques, and address an equivalent formulation with non-overlapping groups, but in higher dimension and with additional constraints. We propose efficient and scalable algorithms exploiting these two strategies, which are significantly faster than alternative approaches. We illustrate these methods with several problems such as CUR matrix factorization, multi-task learning of tree-structured dictionaries, background subtraction in video sequences, image denoising with wavelets, and topographic dictionary learning of natural image patches.

We consider the problem of optimizing the sum of a smooth convex function and a non-smooth convex function using proximal-gradient methods, where an error is present in the calculation of the gradient of the smooth term or in the proximity operator with respect to the non-smooth term. We show that both the basic proximal-gradient method and the accelerated proximal-gradient method achieve the same convergence rate as in the error-free case, provided that the errors decrease at appropriate rates. Using these rates, we perform as well as or better than a carefully chosen fixed error level on a set of structured sparsity problems. 1

We consider the problem of optimizing the sum of a smooth convex function and a non-smooth convex function using proximal-gradient methods, where an error is present in the calculation of the gradient of the smooth term or in the proximity operator with respect to the non-smooth term. We show that both the basic proximal-gradient method and the accelerated proximal-gradient method achieve the same convergence rate as in the error-free case, provided that the errors decrease at appropriate rates. Using these rates, we perform as well as or better than a carefully chosen fixed error level on a set of structured sparsity problems. 1

comparison using robust cross-validation of the performance of eleven computational approaches to real-time electroencephalography (EEG) based mental workload monitoring on Multi-Attribute Task Battery data from eight subjects. We propose a new approach, Overcomplete Spectral Regression, that combines several potentially advantageous attributes and empirically demonstrate its superior performance on these data compared to the ten other CSA methods tested. We discuss results from computational, neuroscience and experimentation points of view. R I.

Recent work in signal processing and statistics have focused on defining new regularization functions, which not only induce sparsity of the solution, but also take into account the structure of the problem. 1–7 We present in this paper a class of convex penalties introduced in the machine learning community, which take the form of a sum of ℓ2- and ℓ∞norms over groups of variables. They extend the classical group-sparsity regularization8–10 in the sense that the groups possibly overlap, allowing more flexibility in the group design. We review efficient optimization methods to deal with the corresponding inverse problems, 11–13 and their application to the problem of learning dictionaries of natural image patches: 14–18 On the one hand, dictionary learning has indeed proven effective for various signal processing tasks. 17, 19 On the other hand, structured sparsity provides a natural framework for modeling dependencies between dictionary elements. We thus consider a structured sparse regularization to learn dictionaries embedded in a particular structure, for instance a tree11 or a two-dimensional grid. 20 In the latter case, the results we obtain are similar to the dictionaries produced by topographic independent component analysis. 21

Linear models have enjoyed great success in structured prediction in NLP. While a lot of progress has been made on efficient training with several loss functions, the problem of endowing learners with a mechanism for feature selection is still unsolved. Common approaches employ ad hoc filtering or L1regularization; both ignore the structure of the feature space, preventing practicioners from encoding structural prior knowledge. We fill this gap by adopting regularizers that promote structured sparsity, along with efficient algorithms to handle them. Experiments on three tasks (chunking, entity recognition, and dependency parsing) show gains in performance, compactness, and model interpretability. 1

Linear models have enjoyed great success in structured prediction in NLP. While a lot of progress has been made on efficient training with several loss functions, the problem of endowing learners with a mechanism for feature selection is still unsolved. Common approaches employ ad hoc filtering or L1regularization; both ignore the structure of the feature space, preventing practicioners from encoding structural prior knowledge. We fill this gap by adopting regularizers that promote structured sparsity, along with efficient algorithms to handle them. Experiments on three tasks (chunking, entity recognition, and dependency parsing) show gains in performance, compactness, and model interpretability. 1

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The partition function plays a key role in probabilistic modeling including conditional random fields, graphical models, and maximum likelihood estimation. To optimize partition functions, this article introduces a quadratic variational upper bound. This inequality facilitates majorization methods: optimization of complicated functions through the iterative solution of simpler sub-problems. Such bounds remain efficient to compute even when the partition function involves a graphical model (with small tree-width) or in latent likelihood settings. For large-scale problems, low-rank versions of the bound are provided and outperform LBFGS as well as first-order methods. Several learning applications are shown and reduce to fast and convergent update rules. Experimental results show advantages over state-of-the-art optimization methods. 1