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172
Alternating direction algorithms for ℓ1problems in compressive sensing
, 2009
"... Abstract. In this paper, we propose and study the use of alternating direction algorithms for several ℓ1norm minimization problems arising from sparse solution recovery in compressive sensing, including the basis pursuit problem, the basispursuit denoising problems of both unconstrained and constr ..."
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Cited by 49 (5 self)
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Abstract. In this paper, we propose and study the use of alternating direction algorithms for several ℓ1norm minimization problems arising from sparse solution recovery in compressive sensing, including the basis pursuit problem, the basispursuit denoising problems of both unconstrained and constrained forms, as well as others. We present and investigate two classes of algorithms derived from either the primal or the dual forms of the ℓ1problems. The construction of the algorithms consists of two main steps: (1) to reformulate an ℓ1problem into one having partially separable objective functions by adding new variables and constraints; and (2) to apply an exact or inexact alternating direction method to the resulting problem. The derived alternating direction algorithms can be regarded as firstorder primaldual algorithms because both primal and dual variables are updated at each and every iteration. Convergence properties of these algorithms are established or restated when they already exist. Extensive numerical results in comparison with several stateoftheart algorithms are given to demonstrate that the proposed algorithms are efficient, stable and robust. Moreover, we present numerical results to emphasize two practically important but perhaps overlooked points. One point is that algorithm speed should always be evaluated relative to appropriate solution accuracy; another is that whenever erroneous measurements possibly exist, the ℓ1norm fidelity should be the fidelity of choice in compressive sensing. Key words. Sparse solution recovery, compressive sensing, ℓ1minimization, primal, dual, alternating direction method
Structured Sparsity through Convex Optimization
, 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 ℓ1norm. In this paper, we consider sit ..."
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Cited by 47 (6 self)
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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 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.
Approximation Accuracy, Gradient Methods, and Error Bound for Structured Convex Optimization
, 2009
"... Convex optimization problems arising in applications, possibly as approximations of intractable problems, are often structured and large scale. When the data are noisy, it is of interest to bound the solution error relative to the (unknown) solution of the original noiseless problem. Related to this ..."
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Cited by 38 (1 self)
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Convex optimization problems arising in applications, possibly as approximations of intractable problems, are often structured and large scale. When the data are noisy, it is of interest to bound the solution error relative to the (unknown) solution of the original noiseless problem. Related to this is an error bound for the linear convergence analysis of firstorder gradient methods for solving these problems. Example applications include compressed sensing, variable selection in regression, TVregularized image denoising, and sensor network localization.
Analysis and generalizations of the linearized Bregman method
 SIAM J. IMAGING SCI
, 2010
"... This paper analyzes and improves the linearized Bregman method for solving the basis pursuit and related sparse optimization problems. The analysis shows that the linearized Bregman method has the exact regularization property; namely, it converges to an exact solution of the basis pursuit problem ..."
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Cited by 36 (9 self)
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This paper analyzes and improves the linearized Bregman method for solving the basis pursuit and related sparse optimization problems. The analysis shows that the linearized Bregman method has the exact regularization property; namely, it converges to an exact solution of the basis pursuit problem whenever its smooth parameter α is greater than a certain value. The analysis is based on showing that the linearized Bregman algorithm is equivalent to gradient descent applied to a certain dual formulation. This result motivates generalizations of the algorithm enabling the use of gradientbased optimization techniques such as line search, Barzilai–Borwein, limited memory BFGS (LBFGS), nonlinear conjugate gradient, and Nesterov’s methods. In the numerical simulations, the two proposed implementations, one using Barzilai–Borwein steps with nonmonotone line search and the other using LBFGS, gave more accurate solutions in much shorter times than the basic implementation of the linearized Bregman method with a socalled kicking technique.
A sparsegroup lasso
 JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS
, 2013
"... For high dimensional supervised learning problems, often using problem specific assumptions can lead to greater accuracy. For problems with grouped covariates, which are believed to have sparse effects both on a group and within group level, we introduce a regularized model for linear regression w ..."
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Cited by 35 (3 self)
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For high dimensional supervised learning problems, often using problem specific assumptions can lead to greater accuracy. For problems with grouped covariates, which are believed to have sparse effects both on a group and within group level, we introduce a regularized model for linear regression with ℓ1 and ℓ2 penalties. We discuss the sparsity and other regularization properties of the optimal fit for this model, and show that it has the desired effect of groupwise and within group sparsity. We propose an algorithm to fit the model via accelerated generalized gradient descent, and extend this model and algorithm to convex loss functions. We also demonstrate the efficacy of our model and the efficiency of our algorithm on simulated data.
Continuous Multiclass Labeling Approaches and Algorithms
 SIAM J. Imag. Sci
, 2011
"... We study convex relaxations of the image labeling problem on a continuous domain with regularizers based on metric interaction potentials. The generic framework ensures existence of minimizers and covers a wide range of relaxations of the originally combinatorial problem. We focus on two specific r ..."
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Cited by 28 (5 self)
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We study convex relaxations of the image labeling problem on a continuous domain with regularizers based on metric interaction potentials. The generic framework ensures existence of minimizers and covers a wide range of relaxations of the originally combinatorial problem. We focus on two specific relaxations that differ in flexibility and simplicity – one can be used to tightly relax any metric interaction potential, while the other one only covers Euclidean metrics but requires less computational effort. For solving the nonsmooth discretized problem, we propose a globally convergent DouglasRachford scheme, and show that a sequence of dual iterates can be recovered in order to provide a posteriori optimality bounds. In a quantitative comparison to two other firstorder methods, the approach shows competitive performance on synthetical and realworld images. By combining the method with an improved binarization technique for nonstandard potentials, we were able to routinely recover discrete solutions within 1%–5 % of the global optimum for the combinatorial image labeling problem. 1 Problem Formulation The multiclass image labeling problem consists in finding, for each pixel x in the image domain Ω ⊆ Rd, a label `(x) ∈ {1,..., l} which assigns one of l class labels to x so that the labeling function ` adheres to some local data fidelity as well as nonlocal spatial coherency constraints. This problem class occurs in many applications, such as segmentation, multiview reconstruction, stitching, and inpainting [PCF06]. We consider the variational formulation inf `:Ω→{1,...,l} f(`), f(`):= Ω s(x, `(x))dx ︸ ︷ ︷ ︸ data term + J(`). ︸ ︷ ︷ ︸ regularizer
Distributed basis pursuit
 IEEE Trans. Sig. Proc
, 2012
"... Abstract—We propose a distributed algorithm for solving the optimization problem Basis Pursuit (BP). BP finds the leastnorm solution of the underdetermined linear system and is used, for example, in compressed sensing for reconstruction. Our algorithm solves BP on a distributed platform such as a s ..."
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Cited by 28 (6 self)
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Abstract—We propose a distributed algorithm for solving the optimization problem Basis Pursuit (BP). BP finds the leastnorm solution of the underdetermined linear system and is used, for example, in compressed sensing for reconstruction. Our algorithm solves BP on a distributed platform such as a sensor network, and is designed to minimize the communication between nodes. The algorithm only requires the network to be connected, has no notion of a central processing node, and no node has access to the entire matrix at any time. We consider two scenarios in which either the columns or the rows of are distributed among the compute nodes. Our algorithm, named DADMM, is a decentralized implementation of the alternating direction method of multipliers. We show through numerical simulation that our algorithm requires considerably less communications between the nodes than the stateoftheart algorithms. Index Terms—Augmented Lagrangian, basis pursuit (BP), distributed optimization, sensor networks.
Efficient Minimization of Decomposable Submodular Functions
"... Many combinatorial problems arising in machine learning can be reduced to the problem of minimizing a submodular function. Submodular functions are a natural discrete analog of convex functions, and can be minimized in strongly polynomial time. Unfortunately, stateoftheart algorithms for general ..."
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Cited by 25 (3 self)
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Many combinatorial problems arising in machine learning can be reduced to the problem of minimizing a submodular function. Submodular functions are a natural discrete analog of convex functions, and can be minimized in strongly polynomial time. Unfortunately, stateoftheart algorithms for general submodular minimization are intractable for larger problems. In this paper, we introduce a novel subclass of submodular minimization problems that we call decomposable. Decomposable submodular functions are those that can be represented as sums of concave functions applied to modular functions. We develop an algorithm, SLG, that can efficiently minimize decomposable submodular functions with tens of thousands of variables. Our algorithm exploits recent results in smoothed convex minimization. We apply SLG to synthetic benchmarks and a joint classificationandsegmentation task, and show that it outperforms the stateoftheart general purpose submodular minimization algorithms by several orders of magnitude. 1
Sparse channel separation using random probes
, 2010
"... This paper considers the problem of estimating the channel response (or Green’s function) between multiple sourcereceiver pairs. Typically, the channel responses are estimated oneatatime: a single source sends out a known probe signal, the receiver measures the probe signal convolved with the ch ..."
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Cited by 24 (6 self)
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This paper considers the problem of estimating the channel response (or Green’s function) between multiple sourcereceiver pairs. Typically, the channel responses are estimated oneatatime: a single source sends out a known probe signal, the receiver measures the probe signal convolved with the channel response, and the responses are recovered using deconvolution. In this paper, we show that if the channel responses are sparse and the probe signals are random, then we can significantly reduce the total amount of time required to probe the channels by activating all of the sources simultaneously. With all sources activated simultaneously, the receiver measures a superposition of all the channel responses convolved with the respective probe signals. Separating this cumulative response into individual channel responses can be posed as a linear inverse problem. We show that channel response separation is possible (and stable) even when the probing signals are relatively short in spite of the corresponding linear system of equations becoming severely underdetermined. We derive a theoretical lower bound on the length of the source signals that guarantees that this separation is possible with high probability. The bound is derived by putting the problem in the context of finding a sparse solution to an underdetermined system of equations, and then using mathematical tools from the theory of compressive sensing. Finally, we discuss some practical applications of these results, which include forward modeling for seismic imaging, channel equalization in multipleinput multipleoutput communication, and increasing the fieldofview in an imaging system by using coded apertures.