Recently, penalties promoting signals that are sparse within and across groups have been proposed. In this letter, we propose a generalization that allows to encode more intricate dependencies within groups. However, this complicates the realization of the threshold function associated with the penalty, which hinders the use of the penalty in energy minimization. We discuss how to sidestep this problem, and demonstrate the use of the modified penalty in an energy minimization formulation for an inverse problem.
The sparsity characteristic sought in this letter is different than that sought in many papers using group-based penalty functions. Specifically, @cite_25 @cite_6 @cite_4 @cite_16 @cite_19 @cite_17 aim to promote signals that can be represented with a few groups, where within groups, the coefficients are less stringently penalized. In contrast, @cite_6 @cite_10 @cite_14 @cite_24 aim a similar characteristic as the proposed penalty. In this collection, the SWAG penalty @cite_24 , which the proposed penalty aims to modify, separates from the rest in that it is a non-convex penalty. In @cite_24 , it was argued that this property reduces the bias in the non-zero estimates produced by the threshold function (see also @cite_28 @cite_13 for related discussions). For a more detailed comparison between the SWAG penalty and the penalties in @cite_6 @cite_10 @cite_14 , we refer to @cite_24 .
@cite_13 @cite_14 @cite_4 @cite_28 @cite_6 @cite_24 @cite_19 @cite_16 @cite_10 @cite_25 @cite_17
1989060270 2096315701 2122950831 1994309289 2529812697 2149967459 2167121755 1551912413 2538773003 2168820925 2111854888
We study the problem of estimating high-dimensional regression models regularized by a structured sparsity-inducing penalty that encodes prior structural information on either the input or output variables. We consider two widely adopted types of penalties of this kind as motivating examples: (1) the general overlapping-group-lasso penalty, generalized from the group-lasso penalty; and (2) the graph-guided-fused-lasso penalty, generalized from the fused-lasso penalty. For both types of penalties, due to their nonseparability and nonsmoothness, developing an efficient optimization method remains a challenging problem. In this paper we propose a general optimization approach, the smoothing proximal gradient (SPG) method, which can solve structured sparse regression problems with any smooth convex loss under a wide spectrum of structured sparsity-inducing penalties. Our approach combines a smoothing technique with an effective proximal gradient method. It achieves a convergence rate significantly faster than the standard first-order methods, subgradient methods, and is much more scalable than the most widely used interior-point methods. The efficiency and scalability of our method are demonstrated on both simulation experiments and real genetic data sets. In this paper we study sparsity-inducing nonconvex penalty functions using Levy processes. We define such a penalty as the Laplace exponent of a subordinator. Accordingly, we propose a novel approach for the construction of sparsity-inducing nonconvex penalties. Particularly, we show that the nonconvex logarithmic (LOG) and exponential (EXP) penalty functions are the Laplace exponents of Gamma and compound Poisson subordinators, respectively. Additionally, we explore the concave conjugate of nonconvex penalties. We find that the LOG and EXP penalties are the concave conjugates of negative Kullback-Leiber (KL) distance functions. Furthermore, the relationship between these two penalties is due to asymmetricity of the KL distance. This paper addresses the problem of sparsity penalized least squares for applications in sparse signal processing, e.g., sparse deconvolution. This paper aims to induce sparsity more strongly than L1 norm regularization, while avoiding non-convex optimization. For this purpose, this paper describes the design and use of non-convex penalty functions (regularizers) constrained so as to ensure the convexity of the total cost function F to be minimized. The method is based on parametric penalty functions, the parameters of which are constrained to ensure convexity of F. It is shown that optimal parameters can be obtained by semidefinite programming (SDP). This maximally sparse convex (MSC) approach yields maximally non-convex sparsity-inducing penalty functions constrained such that the total cost function F is convex. It is demonstrated that iterative MSC (IMSC) can yield solutions substantially more sparse than the standard convex sparsity-inducing approach, i.e., L1 norm minimization. Extracting useful information from high-dimensional data is an important focus of today's statistical research and practice. Penalized loss function minimization has been shown to be effective for this task both theoretically and empirically. With the virtues of both regularization and sparsity, the L 1 -penalized squared error minimization method Lasso has been popular in regression models and beyond. In this paper, we combine different norms including L 1 to form an intelligent penalty in order to add side information to the fitting of a regression or classification model to obtain reasonable estimates. Specifically, we introduce the Composite Absolute Penalties (CAP) family, which allows given grouping and hierarchical relationships between the predictors to be expressed. CAP penalties are built by defining groups and combining the properties of norm penalties at the across-group and within-group levels. Grouped selection occurs for nonoverlapping groups. Hierarchical variable selection is reached by defining groups with particular overlapping patterns. We propose using the BLASSO and cross-validation to compute CAP estimates in general. For a subfamily of CAP estimates involving only the L 1 and L ∞ norms, we introduce the iCAP algorithm to trace the entire regularization path for the grouped selection problem. Within this subfamily, unbiased estimates of the degrees of freedom (df) are derived so that the regularization parameter is selected without cross-validation. CAP is shown to improve on the predictive performance of the LASSO in a series of simulated experiments, including cases with p » n and possibly mis-specified groupings. When the complexity of a model is properly calculated, iCAP is seen to be parsimonious in the experiments. We introduce a new penalty function that promotes signals composed of a small number of active groups, where within each group, only a few high magnitude coefficients are nonzero. We derive the threshold function associated with the proposed penalty and study its properties. We discuss how the proposed penalty threshold function can be useful for signals with isolated nonzeros, such as audio with isolated harmonics along the frequency axis, or reflection functions in exploration seismology where the nonzeros occur on the boundaries of subsoil layers. We demonstrate the use of the proposed penalty threshold functions in a convex denoising and a nonconvex deconvolution formulation. We provide convergent algorithms for both formulations and compare the performance with state-of-the-art methods. This paper considers a sparse signal recovery task in time-varying (time-adaptive) environments. The contribution of the paper to sparsity-aware online learning is threefold; first, a generalized thresholding (GT) operator, which relates to both convex and non-convex penalty functions, is introduced. This operator embodies, in a unified way, the majority of well-known thresholding rules which promote sparsity. Second, a non-convexly constrained, sparsity-promoting, online learning scheme, namely the adaptive projection-based generalized thresholding (APGT), is developed that incorporates the GT operator with a computational complexity that scales linearly to the number of unknowns. Third, the novel family of partially quasi-nonexpansive mappings is introduced as a functional analytic tool for treating the GT operator. By building upon the rich fixed point theory, the previous class of mappings establishes also a link between the GT operator and a union of linear subspaces; a non-convex object which lies at the heart of any sparsity promoting technique, batch or online. Based on this functional analytic framework, a convergence analysis of the APGT is provided. Extensive experiments suggest that the APGT exhibits competitive performance when compared to computationally more demanding alternatives, such as the sparsity-promoting affine projection algorithm (APA)- and recursive least-squares (RLS)-based techniques. This paper is mainly devoted to a precise analysis of what kind of penalties should be used in order to perform model selection via the minimiza- tion of a penalized least-squares type criterion within some general Gaussian framework including the classical ones. As compared to our previous paper on this topic (Birge and Massart in J. Eur. Math. Soc. 3, 203-268 (2001)), more elaborate forms of the penalties are given which are shown to be, in some sense, optimal. We indeed provide more precise upper bounds for the risk of the penalized estimators and lower bounds for the penalty terms, showing that the use of smaller penalties may lead to disastrous results. These lower bounds may also be used to design a practical strategy that allows to estimate the penalty from the data when the amount of noise is unknown. We provide an illustra- tion of the method for the problem of estimating a piecewise constant signal in Gaussian noise when neither the number, nor the location of the change points are known. SUMMARY Group sparsity is one of the important signal priors for regularization of inverse problems. Sparsity with group structure is encountered in numerous applications. However, despite the abundance of sparsity-based adaptive algorithms, attempts at group sparse adaptive methods are very scarce. In this paper, we introduce novel recursive least squares (RLS) adaptive algorithms regularized via penalty functions, which promote group sparsity. We present a new analytic approximation for lp,0 norm to utilize it as a group sparse regularizer. Simulation results confirm the improved performance of the new group sparse algorithms over regular and sparse RLS algorithms when group sparse structure is present. Copyright © 2013 John Wiley & Sons, Ltd. This paper investigates a new learning formulation called dynamic group sparsity. It is a natural extension of the standard sparsity concept in compressive sensing, and is motivated by the observation that in some practical sparse data the nonzero coefficients are often not random but tend to be clustered. Intuitively, better results can be achieved in these cases by reasonably utilizing both clustering and sparsity priors. Motivated by this idea, we have developed a new greedy sparse recovery algorithm, which prunes data residues in the iterative process according to both sparsity and group clustering priors rather than only sparsity as in previous methods. The proposed algorithm can recover stably sparse data with clustering trends using far fewer measurements and computations than current state-of-the-art algorithms with provable guarantees. Moreover, our algorithm can adaptively learn the dynamic group structure and the sparsity number if they are not available in the practical applications. We have applied the algorithm to sparse recovery and background subtraction in videos. Numerous experiments with improved performance over previous methods further validate our theoretical proofs and the effectiveness of the proposed algorithm. We address the problem of learning structured unsupervised models with moment sparsity typical in many natural language induction tasks. For example, in unsupervised part-of-speech (POS) induction using hidden Markov models, we introduce a bias for words to be labeled by a small number of tags. In order to express this bias of posterior sparsity as opposed to parametric sparsity, we extend the posterior regularization framework [7]. We evaluate our methods on three languages — English, Bulgarian and Portuguese — showing consistent and significant accuracy improvement over EM-trained HMMs, and HMMs with sparsity-inducing Dirichlet priors trained by variational EM. We increase accuracy with respect to EM by 2.3 -6.5 in a purely unsupervised setting as well as in a weakly-supervised setting where the closed-class words are provided. Finally, we show improvements using our method when using the induced clusters as features of a discriminative model in a semi-supervised setting. Sparsity of representations of signals has been shown to be a key concept of fundamental importance in fields such as blind source separation, compression, sampling and signal analysis. The aim of this paper is to compare several commonly-used sparsity measures based on intuitive attributes. Intuitively, a sparse representation is one in which a small number of coefficients contain a large proportion of the energy. In this paper, six properties are discussed: (Robin Hood, Scaling, Rising Tide, Cloning, Bill Gates, and Babies), each of which a sparsity measure should have. The main contributions of this paper are the proofs and the associated summary table which classify commonly-used sparsity measures based on whether or not they satisfy these six propositions. Only two of these measures satisfy all six: the pq-mean with p les 1, q > 1 and the Gini index.