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79
Group Lasso with Overlap and Graph Lasso
"... We propose a new penalty function which, when used as regularization for empirical risk minimization procedures, leads to sparse estimators. The support of the sparse vector is typically a union of potentially overlapping groups of covariates defined a priori, or a set of covariates which tend to be ..."
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Cited by 112 (12 self)
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We propose a new penalty function which, when used as regularization for empirical risk minimization procedures, leads to sparse estimators. The support of the sparse vector is typically a union of potentially overlapping groups of covariates defined a priori, or a set of covariates which tend to be connected to each other when a graph of covariates is given. We study theoretical properties of the estimator, and illustrate its behavior on simulated and breast cancer gene expression data. 1.
Structured variable selection with sparsityinducing norms
, 904
"... We consider the empirical risk minimization problem for linear supervised learning, with regularization by structured sparsityinducing norms. These are defined as sums of Euclidean norms on certain subsets of variables, extending the usual ℓ1norm and the group ℓ1norm by allowing the subsets to ov ..."
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Cited by 97 (15 self)
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We consider the empirical risk minimization problem for linear supervised learning, with regularization by structured sparsityinducing norms. These are defined as sums of Euclidean norms on certain subsets of variables, extending the usual ℓ1norm and the group ℓ1norm by allowing the subsets to overlap. This leads to a specific set of allowed nonzero patterns for the solutions of such problems. We first explore the relationship between the groups defining the norm and the resulting nonzero patterns, providing both forward and backward algorithms to go back and forth from groups to patterns. This allows the design of norms adapted to specific prior knowledge expressed in terms of nonzero patterns. We also present an efficient active set algorithm, and analyze the consistency of variable selection for leastsquares linear regression in low and highdimensional settings.
Online learning for matrix factorization and sparse coding
"... Sparse coding—that is, modelling data vectors as sparse linear combinations of basis elements—is widely used in machine learning, neuroscience, signal processing, and statistics. This paper focuses on the largescale matrix factorization problem that consists of learning the basis set, adapting it t ..."
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Cited by 97 (18 self)
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Sparse coding—that is, modelling data vectors as sparse linear combinations of basis elements—is widely used in machine learning, neuroscience, signal processing, and statistics. This paper focuses on the largescale matrix factorization problem that consists of learning the basis set, adapting it to specific data. Variations of this problem include dictionary learning in signal processing, nonnegative matrix factorization and sparse principal component analysis. In this paper, we propose to address these tasks with a new online optimization algorithm, based on stochastic approximations, which scales up gracefully to large datasets with millions of training samples, and extends naturally to various matrix factorization formulations, making it suitable for a wide range of learning problems. A proof of convergence is presented, along with experiments with natural images and genomic data demonstrating that it leads to stateoftheart performance in terms of speed and optimization for both small and large datasets.
TreeGuided Group Lasso for MultiTask Regression with Structured Sparsity
"... We consider the problem of learning a sparse multitask regression, where the structure in the outputs can be represented as a tree with leaf nodes as outputs and internal nodes as clusters of the outputs at multiple granularity. Our goal is to recover the common set of relevant inputs for each outp ..."
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Cited by 63 (9 self)
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We consider the problem of learning a sparse multitask regression, where the structure in the outputs can be represented as a tree with leaf nodes as outputs and internal nodes as clusters of the outputs at multiple granularity. Our goal is to recover the common set of relevant inputs for each output cluster. Assuming that the tree structure is available as prior knowledge, we formulate this problem as a new multitask regularized regression called treeguided group lasso. Our structured regularization is based on a grouplasso penalty, where groups are defined with respect to the tree structure. We describe a systematic weighting scheme for the groups in the penalty such that each output variable is penalized in a balanced manner even if the groups overlap. We present an efficient optimization method that can handle a largescale problem. Using simulated and yeast datasets, we demonstrate that our method shows a superior performance in terms of both prediction errors and recovery of true sparsity patterns compared to other methods for multitask learning. 1.
A simpler approach to matrix completion
 the Journal of Machine Learning Research
"... This paper provides the best bounds to date on the number of randomly sampled entries required to reconstruct an unknown low rank matrix. These results improve on prior work by Candès and Recht [4], Candès and Tao [7], and Keshavan, Montanari, and Oh [18]. The reconstruction is accomplished by minim ..."
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Cited by 58 (3 self)
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This paper provides the best bounds to date on the number of randomly sampled entries required to reconstruct an unknown low rank matrix. These results improve on prior work by Candès and Recht [4], Candès and Tao [7], and Keshavan, Montanari, and Oh [18]. The reconstruction is accomplished by minimizing the nuclear norm, or sum of the singular values, of the hidden matrix subject to agreement with the provided entries. If the underlying matrix satisfies a certain incoherence condition, then the number of entries required is equal to a quadratic logarithmic factor times the number of parameters in the singular value decomposition. The proof of this assertion is short, self contained, and uses very elementary analysis. The novel techniques herein are based on recent work in quantum information theory.
An Accelerated Gradient Method for Trace Norm Minimization
"... We consider the minimization of a smooth loss function regularized by the trace norm of the matrix variable. Such formulation finds applications in many machine learning tasks including multitask learning, matrix classification, and matrix completion. The standard semidefinite programming formulati ..."
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Cited by 56 (5 self)
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We consider the minimization of a smooth loss function regularized by the trace norm of the matrix variable. Such formulation finds applications in many machine learning tasks including multitask learning, matrix classification, and matrix completion. The standard semidefinite programming formulation for this problem is computationally expensive. In addition, due to the nonsmooth nature of the trace norm, the optimal firstorder blackbox method for solving such class of problems converges as O ( 1 √), where k is the k iteration counter. In this paper, we exploit the special structure of the trace norm, based on which we propose an extended gradient algorithm that converges as O ( 1 k). We further propose an accelerated gradient algorithm, which achieves the optimal convergence rate of O ( 1 k 2) for smooth problems. Experiments on multitask learning problems demonstrate the efficiency of the proposed algorithms. 1.
Structured Sparse Principal Component Analysis
, 2009
"... We present an extension of sparse PCA, or sparse dictionary learning, where the sparsity patterns of all dictionary elements are structured and constrained to belong to a prespecified set of shapes. This structured sparse PCA is based on a structured regularization recently introduced by [1]. While ..."
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Cited by 34 (12 self)
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We present an extension of sparse PCA, or sparse dictionary learning, where the sparsity patterns of all dictionary elements are structured and constrained to belong to a prespecified set of shapes. This structured sparse PCA is based on a structured regularization recently introduced by [1]. While classical sparse priors only deal with cardinality, the regularization we use encodes higherorder information about the data. We propose an efficient and simple optimization procedure to solve this problem. Experiments with two practical tasks, face recognition and the study of the dynamics of a protein complex, demonstrate the benefits of the proposed structured approach over unstructured approaches. 1
SUPPORT UNION RECOVERY IN HIGHDIMENSIONAL MULTIVARIATE REGRESSION
 SUBMITTED TO THE ANNALS OF STATISTICS
, 2010
"... In multivariate regression, a Kdimensional response vector is regressed upon a common set of p covariates, with a matrix B ∗ ∈ R p×K of regression coefficients. We study the behavior of the multivariate group Lasso, in which block regularization based on the ℓ1/ℓ2 norm is used for support union re ..."
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Cited by 26 (1 self)
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In multivariate regression, a Kdimensional response vector is regressed upon a common set of p covariates, with a matrix B ∗ ∈ R p×K of regression coefficients. We study the behavior of the multivariate group Lasso, in which block regularization based on the ℓ1/ℓ2 norm is used for support union recovery, or recovery of the set of s rows for which B ∗ is nonzero. Under highdimensional scaling, we show that the multivariate group Lasso exhibits a threshold for the recovery of the exact row pattern with high probability over the random design and noise that is specified by the sample complexity parameter θ(n, p, s) : = n/[2ψ(B ∗ ) log(p − s)]. Here n is the sample size, and ψ(B ∗ ) is a sparsityoverlap function measuring a combination of the sparsities and overlaps of the Kregression coefficient vectors that constitute the model. We prove that the multivariate group Lasso succeeds for problem sequences (n, p, s) such that θ(n, p, s) exceeds a critical level θu, and fails for sequences such that θ(n, p, s) lies below a critical level θℓ. For the special case of the standard Gaussian ensemble, we show that θℓ = θu so that the characterization is sharp. The sparsityoverlap function ψ(B ∗ ) reveals that, if the design is uncorrelated on the active rows, ℓ1/ℓ2 regularization for multivariate regression never harms performance relative to an ordinary Lasso approach, and can yield substantial improvements in sample complexity (up to a factor of K) when the coefficient vectors are suitably orthogonal. For more general designs, it is possible for the ordinary Lasso to outperform the multivariate group Lasso. We complement our analysis with simulations that demonstrate the sharpness of our theoretical results, even for relatively small problems.