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14
Structured sparsityinducing norms through submodular functions
 IN ADVANCES IN NEURAL INFORMATION PROCESSING SYSTEMS
, 2010
"... 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 turnedinto a convex optimization problem byreplacing the cardinality function by its convex en ..."
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Cited by 61 (12 self)
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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 turnedinto a convex optimization problem byreplacing the cardinality function by its convex envelope (tightest convex lower bound), in this case the ℓ1norm. In this paper, we investigate more general setfunctions than the cardinality, that may incorporate prior knowledge or structural constraints which are common in many applications: namely, we show that for nonincreasing submodular setfunctions, the corresponding convex envelope can be obtained from its Lovász 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 highdimensional inference). By selecting specific submodular functions, we can give a new interpretation to known norms, such as those based on rankstatistics or grouped norms with potentially overlapping groups; we also define new norms, in particular ones that can be used as nonfactorial priors for supervised learning.
Structured Sparsity through Convex Optimization
"... Abstract. Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. While naturally cast as a combinatorial optimization problem, variable or feature selection admits a convex relaxation through the regularization by the ℓ1norm. In this paper, we cons ..."
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Cited by 48 (7 self)
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Abstract. Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. While naturally cast as a combinatorial optimization problem, variable or feature selection admits a convex relaxation through the regularization by the ℓ1norm. In this paper, we consider situations where we are not only interested in sparsity, but where some structural prior knowledge is available as well. We show that the ℓ1norm can then be extended to structured norms built on either disjoint or overlapping groups of variables, leading to a flexible framework that can deal with various structures. We present applications to unsupervised learning, for structured sparse principal component analysis and hierarchical dictionary learning, and to supervised learning in the context of nonlinear variable selection. Key words and phrases: Sparsity, convex optimization. 1.
Supervised Feature Selection in Graphs with Path Coding Penalties and Network Flows
, 2011
"... We consider supervised learning problems where the features are embedded in a graph, such as gene expressions in a gene network. In this context, it is of much interest to take into account the problem structure, and automatically select a subgraph with a small number of connected components. By exp ..."
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Cited by 9 (3 self)
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We consider supervised learning problems where the features are embedded in a graph, such as gene expressions in a gene network. In this context, it is of much interest to take into account the problem structure, and automatically select a subgraph with a small number of connected components. By exploiting prior knowledge, one can indeed improve the prediction performance and/or obtain better interpretable results. Regularization or penalty functions for selecting features in graphs have recently been proposed but they raise new algorithmic challenges. For example, they typically require solving a combinatorially hard selection problem among all connected subgraphs. In this paper, we propose computationally feasible strategies to select a sparse and “well connected” subset of features sitting on a directed acyclic graph (DAG). We introduce structured sparsity penalties over paths on a DAG called “path coding ” penalties. Unlike existing regularization functions, path coding penalties can both model long range interactions between features in the graph and be tractable. The penalties and their proximal operators involve path selection problems, which we efficiently solve by leveraging network flow optimization. We experimentally show on synthetic, image, and genomic data that our approach is scalable and lead to more connected subgraphs than other regularization functions for graphs.
Convex relaxations of structured matrix factorizations
, 2013
"... We consider the factorization of a rectangular matrix X into a positive linear combination of rankone factors of the form uv ⊤ , where u and v belongs to certain sets U and V, that may encode specific structures regarding the factors, such as positivity or sparsity. In this paper, we show that comp ..."
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Cited by 7 (3 self)
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We consider the factorization of a rectangular matrix X into a positive linear combination of rankone factors of the form uv ⊤ , where u and v belongs to certain sets U and V, that may encode specific structures regarding the factors, such as positivity or sparsity. In this paper, we show that computing the optimal decomposition is equivalent to computing a certain gauge function of X and we provide a detailed analysis of these gauge functions and their polars. Since these gaugefunctions are typically hard to compute, we present semidefinite relaxations and several algorithms that may recover approximate decompositions with approximation guarantees. We illustrate our results with simulations on finding decompositions with elements in {0,1}. As side contributions, we present a detailed analysis of variational quadratic representations of norms as well as a new iterative basis pursuit algorithm that can deal with inexact firstorder oracles. 1
New perspectives on ksupport and cluster norms. See http: //arxiv.org/abs/1403.1481
, 2014
"... The ksupport norm is a regularizer which has been successfully applied to sparse vector prediction problems. We show that it belongs to a general class of norms which can be formulated as a parameterized infimum over quadratics. We further extend the ksupport norm to matrices, and we observe that ..."
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Cited by 1 (0 self)
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The ksupport norm is a regularizer which has been successfully applied to sparse vector prediction problems. We show that it belongs to a general class of norms which can be formulated as a parameterized infimum over quadratics. We further extend the ksupport norm to matrices, and we observe that it is a special case of the matrix cluster norm. Using this formulation we derive an efficient algorithm to compute the proximity operator of both norms. This improves upon the standard algorithm for the ksupport norm and allows us to apply proximal gradient methods to the cluster norm. We also describe how to solve regularization problems which employ centered versions of these norms. Finally, we apply the matrix regularizers to different matrix completion and multitask learning datasets. Our results indicate that the spectral ksupport norm and the cluster norm give state of the art performance on these problems, significantly outperforming trace norm and elastic net penalties.
Learning the Structure for Structured Sparsity
"... Abstract—Structured sparsity has recently emerged in statistics, machine learning and signal processing as a promising paradigm for learning in highdimensional settings. All existing methods for learning under the assumption of structured sparsity rely on prior knowledge on how to weight (or how to ..."
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Abstract—Structured sparsity has recently emerged in statistics, machine learning and signal processing as a promising paradigm for learning in highdimensional settings. All existing methods for learning under the assumption of structured sparsity rely on prior knowledge on how to weight (or how to penalize) individual subsets of variables during the subset selection process, which is not available in general. Inferring group weights from data is a key open research problem in structured sparsity. In this paper, we propose a Bayesian approach to the problem of group weight learning. We model the group weights as hyperparameters of heavytailed priors on groups of variables and derive an approximate inference scheme to infer these hyperparameters. We empirically show that we are able to recover the model hyperparameters when the data are generated from the model, and we demonstrate the utility of learning weights in synthetic and real denoising problems. Index Terms — Structured sparsity, probabilistic modeling, Bayesian statistics, superGaussian prior, Gaussian scale mixture,
Structured sparsity through convex optimization
 SUBMITTED TO THE STATISTICAL SCIENCE
, 2012
"... Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. While naturally cast as a combinatorial optimization problem, variable or feature selection admits a convex relaxation through the regularization by the `1norm. In this paper, we consider sit ..."
Abstract
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Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. While naturally cast as a combinatorial optimization problem, variable or feature selection admits a convex relaxation through the regularization by the `1norm. In this paper, we consider situations where we are not only interested in sparsity, but where some structural prior knowledge is available as well. We show that the `1norm can then be extended to structured norms built on either disjoint or overlapping groups of variables, leading to a flexible framework that can deal with various structures. We present applications to unsupervised learning, for structured sparse principal component analysis and hierarchical dictionary learning, and to supervised learning in the context of nonlinear variable selection.