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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 553 (26 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.
The Augmented Lagrange Multiplier Method for Exact Recovery of Corrupted LowRank Matrices
, 2009
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Graph implementations for nonsmooth convex programs
 Recent Advances in Learning and Control, Lecture Notes in Control and Information Sciences
, 2008
"... Summary. We describe graph implementations, a generic method for representing a convex function via its epigraph, described in a disciplined convex programming framework. This simple and natural idea allows a very wide variety of smooth and nonsmooth convex programs to be easily specified and effi ..."
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Cited by 248 (7 self)
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Summary. We describe graph implementations, a generic method for representing a convex function via its epigraph, described in a disciplined convex programming framework. This simple and natural idea allows a very wide variety of smooth and nonsmooth convex programs to be easily specified and efficiently solved, using interiorpoint methods for smooth or cone convex programs. Key words: Convex optimization, nonsmooth optimization, disciplined convex programming, optimization modeling languages, semidefinite program
HighResolution Radar via Compressed Sensing
, 2008
"... A stylized compressed sensing radar is proposed in which the timefrequency plane is discretized into an N ×N grid. Assuming the number of targets K is small (i.e., K ≪ N 2), then we can transmit a sufficiently “incoherent ” pulse and employ the techniques of compressed sensing to reconstruct the ta ..."
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Cited by 156 (9 self)
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A stylized compressed sensing radar is proposed in which the timefrequency plane is discretized into an N ×N grid. Assuming the number of targets K is small (i.e., K ≪ N 2), then we can transmit a sufficiently “incoherent ” pulse and employ the techniques of compressed sensing to reconstruct the target scene. A theoretical upper bound on the sparsity K is presented. Numerical simulations verify that even better performance can be achieved in practice. This novel compressed sensing approach offers great potential for better resolution over classical radar.
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 124 (7 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.
Fast Model Predictive Control Using Online Optimization
, 2008
"... A widely recognized shortcoming of model predictive control (MPC) is that it can usually only be used in applications with slow dynamics, where the sample time is measured in seconds or minutes. A well known technique for implementing fast MPC is to compute the entire control law offline, in which c ..."
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Cited by 110 (23 self)
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A widely recognized shortcoming of model predictive control (MPC) is that it can usually only be used in applications with slow dynamics, where the sample time is measured in seconds or minutes. A well known technique for implementing fast MPC is to compute the entire control law offline, in which case the online controller can be implemented as a lookup table. This method works well for systems with small state and input dimensions (say, no more than 5), and short time horizons. In this paper we describe a collection of methods for improving the speed of MPC, using online optimization. These custom methods, which exploit the particular structure of the MPC problem, can compute the control action on the order of 100 times faster than a method that uses a generic optimizer. As an example, our method computes the control actions for a problem with 12 states, 3 controls, and horizon of 30 time steps (which entails solving a quadratic program with 450 variables and 1260 constraints) in around 5msec, allowing MPC to be carried out at 200Hz. 1
Spectral Regularization Algorithms for Learning Large Incomplete Matrices
, 2009
"... We use convex relaxation techniques to provide a sequence of regularized lowrank solutions for largescale matrix completion problems. Using the nuclear norm as a regularizer, we provide a simple and very efficient convex algorithm for minimizing the reconstruction error subject to a bound on the n ..."
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Cited by 104 (5 self)
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We use convex relaxation techniques to provide a sequence of regularized lowrank solutions for largescale matrix completion problems. Using the nuclear norm as a regularizer, we provide a simple and very efficient convex algorithm for minimizing the reconstruction error subject to a bound on the nuclear norm. Our algorithm SoftImpute iteratively replaces the missing elements with those obtained from a softthresholded SVD. With warm starts this allows us to efficiently compute an entire regularization path of solutions on a grid of values of the regularization parameter. The computationally intensive part of our algorithm is in computing a lowrank SVD of a dense matrix. Exploiting the problem structure, we show that the task can be performed with a complexity linear in the matrix dimensions. Our semidefiniteprogramming algorithm is readily scalable to large matrices: for example it can obtain a rank80 approximation of a 10 6 × 10 6 incomplete matrix with 10 5 observed entries in 2.5 hours, and can fit a rank 40 approximation to the full Netflix training set in 6.6 hours. Our methods show very good performance both in training and test error when compared to other competitive stateofthe art techniques. 1.
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 93 (9 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.
Optimal beamforming for twoway multiantenna relay channel with analogue network coding
 IEEE J. Sel. Areas Commun
, 2009
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Stability results for random sampling of sparse trigonometric polynomials
, 2006
"... Recently, it has been observed that a sparse trigonometric polynomial, i.e. having only a small number of nonzero coefficients, can be reconstructed exactly from a small number of random samples using Basis Pursuit (BP) and Orthogonal Matching Pursuit (OMP). In the present article it is shown that ..."
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Cited by 65 (18 self)
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Recently, it has been observed that a sparse trigonometric polynomial, i.e. having only a small number of nonzero coefficients, can be reconstructed exactly from a small number of random samples using Basis Pursuit (BP) and Orthogonal Matching Pursuit (OMP). In the present article it is shown that recovery both by a BP variant and by OMP is stable under perturbation of the samples values by noise. For BP in addition, the stability result is extended to (nonsparse) trigonometric polynomials that can be wellapproximated by sparse ones. The theoretical findings are illustrated by numerical experiments. Key Words: random sampling, trigonometric polynomials, Orthogonal Matching Pursuit, Basis Pursuit, compressed sensing, stability under noise, fast Fourier transform, nonequispaced