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40
Robust Principal Component Analysis?
, 2009
"... This paper is about a curious phenomenon. Suppose we have a data matrix, which is the superposition of a lowrank component and a sparse component. Can we recover each component individually? We prove that under some suitable assumptions, it is possible to recover both the lowrank and the sparse co ..."
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Cited by 138 (6 self)
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This paper is about a curious phenomenon. Suppose we have a data matrix, which is the superposition of a lowrank component and a sparse component. Can we recover each component individually? We prove that under some suitable assumptions, it is possible to recover both the lowrank and the sparse components exactly by solving a very convenient convex program called Principal Component Pursuit; among all feasible decompositions, simply minimize a weighted combination of the nuclear norm and of the ℓ1 norm. This suggests the possibility of a principled approach to robust principal component analysis since our methodology and results assert that one can recover the principal components of a data matrix even though a positive fraction of its entries are arbitrarily corrupted. This extends to the situation where a fraction of the entries are missing as well. We discuss an algorithm for solving this optimization problem, and present applications in the area of video surveillance, where our methodology allows for the detection of objects in a cluttered background, and in the area of face recognition, where it offers a principled way of removing shadows and specularities in images of faces.
NESTA: A Fast and Accurate FirstOrder Method for Sparse Recovery
, 2009
"... Accurate signal recovery or image reconstruction from indirect and possibly undersampled data is a topic of considerable interest; for example, the literature in the recent field of compressed sensing is already quite immense. Inspired by recent breakthroughs in the development of novel firstorder ..."
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Cited by 73 (2 self)
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Accurate signal recovery or image reconstruction from indirect and possibly undersampled data is a topic of considerable interest; for example, the literature in the recent field of compressed sensing is already quite immense. Inspired by recent breakthroughs in the development of novel firstorder methods in convex optimization, most notably Nesterov’s smoothing technique, this paper introduces a fast and accurate algorithm for solving common recovery problems in signal processing. In the spirit of Nesterov’s work, one of the key ideas of this algorithm is a subtle averaging of sequences of iterates, which has been shown to improve the convergence properties of standard gradientdescent algorithms. This paper demonstrates that this approach is ideally suited for solving largescale compressed sensing reconstruction problems as 1) it is computationally efficient, 2) it is accurate and returns solutions with several correct digits, 3) it is flexible and amenable to many kinds of reconstruction problems, and 4) it is robust in the sense that its excellent performance across a wide range of problems does not depend on the fine tuning of several parameters. Comprehensive numerical experiments on realistic signals exhibiting a large dynamic range show that this algorithm compares favorably with recently proposed stateoftheart methods. We also apply the algorithm to solve other problems for which there are fewer alternatives, such as totalvariation minimization, and
Beyond Nyquist: Efficient Sampling of Sparse Bandlimited Signals
, 2009
"... Wideband analog signals push contemporary analogtodigital conversion systems to their performance limits. In many applications, however, sampling at the Nyquist rate is inefficient because the signals of interest contain only a small number of significant frequencies relative to the bandlimit, alt ..."
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Cited by 68 (13 self)
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Wideband analog signals push contemporary analogtodigital conversion systems to their performance limits. In many applications, however, sampling at the Nyquist rate is inefficient because the signals of interest contain only a small number of significant frequencies relative to the bandlimit, although the locations of the frequencies may not be known a priori. For this type of sparse signal, other sampling strategies are possible. This paper describes a new type of data acquisition system, called a random demodulator, that is constructed from robust, readily available components. Let K denote the total number of frequencies in the signal, and let W denote its bandlimit in Hz. Simulations suggest that the random demodulator requires just O(K log(W/K)) samples per second to stably reconstruct the signal. This sampling rate is exponentially lower than the Nyquist rate of W Hz. In contrast with Nyquist sampling, one must use nonlinear methods, such as convex programming, to recover the signal from the samples taken by the random demodulator. This paper provides a detailed theoretical analysis of the system’s performance that supports the empirical observations.
Bregman iterative algorithms for ℓ1minimization with applications to compressed sensing
 SIAM J. Imaging Sci
, 2008
"... Abstract. We propose simple and extremely efficient methods for solving the basis pursuit problem min{‖u‖1: Au = f,u ∈ R n}, which is used in compressed sensing. Our methods are based on Bregman iterative regularization, and they give a very accurate solution after solving only a very small number o ..."
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Cited by 60 (13 self)
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Abstract. We propose simple and extremely efficient methods for solving the basis pursuit problem min{‖u‖1: Au = f,u ∈ R n}, which is used in compressed sensing. Our methods are based on Bregman iterative regularization, and they give a very accurate solution after solving only a very small number of 1 instances of the unconstrained problem minu∈Rn μ‖u‖1 + 2 ‖Au−fk ‖ 2 2 for given matrix A and vector f k. We show analytically that this iterative approach yields exact solutions in a finite number of steps and present numerical results that demonstrate that as few as two to six iterations are sufficient in most cases. Our approach is especially useful for many compressed sensing applications where matrixvector operations involving A and A ⊤ can be computed by fast transforms. Utilizing a fast fixedpoint continuation solver that is based solely on such operations for solving the above unconstrained subproblem, we were able to quickly solve huge instances of compressed sensing problems on a standard PC.
Robust principal component analysis: Exact recovery of corrupted lowrank matrices via convex optimization
 Advances in Neural Information Processing Systems 22
, 2009
"... The supplementary material to the NIPS version of this paper [4] contains a critical error, which was discovered several days before the conference. Unfortunately, it was too late to withdraw the paper from the proceedings. Fortunately, since that time, a correct analysis of the proposed convex prog ..."
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Cited by 43 (3 self)
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The supplementary material to the NIPS version of this paper [4] contains a critical error, which was discovered several days before the conference. Unfortunately, it was too late to withdraw the paper from the proceedings. Fortunately, since that time, a correct analysis of the proposed convex programming relaxation has been developed by Emmanuel Candes of Stanford University. That analysis is reported in a joint paper, Robust Principal Component Analysis? by Emmanuel Candes, Xiaodong Li, Yi Ma and John Wright,
Fast convex optimization algorithms for exact recovery of a corrupted lowrank matrix
 In Intl. Workshop on Comp. Adv. in MultiSensor Adapt. Processing, Aruba, Dutch Antilles
, 2009
"... Abstract. This paper studies algorithms for solving the problem of recovering a lowrank matrix with a fraction of its entries arbitrarily corrupted. This problem can be viewed as a robust version of classical PCA, and arises in a number of application domains, including image processing, web data r ..."
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Cited by 33 (6 self)
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Abstract. This paper studies algorithms for solving the problem of recovering a lowrank matrix with a fraction of its entries arbitrarily corrupted. This problem can be viewed as a robust version of classical PCA, and arises in a number of application domains, including image processing, web data ranking, and bioinformatic data analysis. It was recently shown that under surprisingly broad conditions, it can be exactly solved via a convex programming surrogate that combines nuclear norm minimization and ℓ1norm minimization. This paper develops and compares two complementary approaches for solving this convex program. The first is an accelerated proximal gradient algorithm directly applied to the primal; while the second is a gradient algorithm applied to the dual problem. Both are several orders of magnitude faster than the previous stateoftheart algorithm for this problem, which was based on iterative thresholding. Simulations demonstrate the performance improvement that can be obtained via these two algorithms, and clarify their relative merits.
Templates for Convex Cone Problems with Applications to Sparse Signal Recovery
, 2010
"... This paper develops a general framework for solving a variety of convex cone problems that frequently arise in signal processing, machine learning, statistics, and other fields. The approach works as follows: first, determine a conic formulation of the problem; second, determine its dual; third, app ..."
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Cited by 31 (3 self)
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This paper develops a general framework for solving a variety of convex cone problems that frequently arise in signal processing, machine learning, statistics, and other fields. The approach works as follows: first, determine a conic formulation of the problem; second, determine its dual; third, apply smoothing; and fourth, solve using an optimal firstorder method. A merit of this approach is its flexibility: for example, all compressed sensing problems can be solved via this approach. These include models with objective functionals such as the totalvariation norm, ‖W x‖1 where W is arbitrary, or a combination thereof. In addition, the paper also introduces a number of technical contributions such as a novel continuation scheme, a novel approach for controlling the step size, and some new results showing that the smooth and unsmoothed problems are sometimes formally equivalent. Combined with our framework, these lead to novel, stable and computationally efficient algorithms. For instance, our general implementation is competitive with stateoftheart methods for solving intensively studied problems such as the LASSO. Further, numerical experiments show that one can solve the Dantzig selector problem, for which no efficient largescale solvers exist, in a few hundred iterations. Finally, the paper is accompanied with a software release. This software is not a single, monolithic solver; rather, it is a suite of programs and routines designed to serve as building blocks for constructing complete algorithms. Keywords. Optimal firstorder methods, Nesterov’s accelerated descent algorithms, proximal algorithms, conic duality, smoothing by conjugation, the Dantzig selector, the LASSO, nuclearnorm minimization.
MoreauYosida Regularization for Grouped Tree Structure Learning
"... We consider the tree structured group Lasso where the structure over the features can be represented as a tree with leaf nodes as features and internal nodes as clusters of the features. The structured regularization with a predefined tree structure is based on a groupLasso penalty, where one grou ..."
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Cited by 25 (4 self)
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We consider the tree structured group Lasso where the structure over the features can be represented as a tree with leaf nodes as features and internal nodes as clusters of the features. The structured regularization with a predefined tree structure is based on a groupLasso penalty, where one group is defined for each node in the tree. Such a regularization can help uncover the structured sparsity, which is desirable for applications with some meaningful tree structures on the features. However, the tree structured group Lasso is challenging to solve due to the complex regularization. In this paper, we develop an efficient algorithm for the tree structured group Lasso. One of the key steps in the proposed algorithm is to solve the MoreauYosida regularization associated with the grouped tree structure. The main technical contributions of this paper include (1) we show that the associated MoreauYosida regularization admits an analytical solution, and (2) we develop an efficient algorithm for determining the effective interval for the regularization parameter. Our experimental results on the AR and JAFFE face data sets demonstrate the efficiency and effectiveness of the proposed algorithm. 1
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 23 (2 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
2010 Analysis and generalizations of the linearized Bregman method
 SIAM J. Imaging Sci
"... Abstract. 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 ..."
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Cited by 19 (5 self)
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Abstract. 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. Key words. Bregman, linearized Bregman, compressed sensing, ℓ1minimization, basis pursuit