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23
Sparse representation of a polytope and recovery of a sparse signals and lowrank matrices
 IEEE Transactions on Information Theory
, 2014
"... This paper considers compressed sensing and affine rank minimization in both noiseless and noisy cases and establishes sharp restricted isometry conditions for sparse signal and lowrank matrix recovery. The analysis relies on a key technical tool which represents points in a polytope by convex comb ..."
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This paper considers compressed sensing and affine rank minimization in both noiseless and noisy cases and establishes sharp restricted isometry conditions for sparse signal and lowrank matrix recovery. The analysis relies on a key technical tool which represents points in a polytope by convex combinations of sparse vectors. The technique is elementary while leads to sharp results. It is shown that for any given constant t ≥ 4/3, in compressed sensing δAtk < (t − 1)/t guarantees the exact recovery of all k sparse signals in the noiseless case through the constrained `1 minimization, and similarly in affine rank minimization δ M tr < (t − 1)/t ensures the exact reconstruction of all matrices with rank at most r in the noiseless case via the constrained nuclear norm minimization. Moreover, for any > 0, δAtk < t−1 t + is not sufficient to guarantee the exact recovery of all ksparse signals for large k. Similar result also holds for matrix recovery. In addition, the conditions δAtk < (t − 1)/t and δMtr < (t − 1)/t are also shown to be sufficient respectively for stable recovery of approximately sparse signals and lowrank matrices in the noisy case.
Stable Restoration and Separation of Approximately Sparse Signals
"... This paper develops new theory and algorithms to recover signals that are approximately sparse in some general (i.e., basis, frame, overcomplete, or incomplete) dictionary but corrupted by a combination of measurement noise and interference having a sparse representation in a second general diction ..."
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This paper develops new theory and algorithms to recover signals that are approximately sparse in some general (i.e., basis, frame, overcomplete, or incomplete) dictionary but corrupted by a combination of measurement noise and interference having a sparse representation in a second general dictionary. Particular applications covered by our framework include the restoration of signals impaired by impulse noise, narrowband interference, or saturation, as well as image inpainting, superresolution, and signal separation. We develop efficient recovery algorithms and deterministic conditions that guarantee stable restoration and separation. Two application examples demonstrate the efficacy of our approach.
Compressed sensing and affine rank minimization under restricted isometry
 IEEE Trans. Signal Process
, 2013
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ROP: Matrix recovery via rankone projections
 The Annals of Statistics
"... Estimation of lowrank matrices is of significant interest in a range of contemporary applications. In this paper, we introduce a rankone projection model for lowrank matrix recovery and propose a constrained nuclear norm minimization method for stable recovery of lowrank matrices in the noisy ca ..."
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Cited by 5 (0 self)
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Estimation of lowrank matrices is of significant interest in a range of contemporary applications. In this paper, we introduce a rankone projection model for lowrank matrix recovery and propose a constrained nuclear norm minimization method for stable recovery of lowrank matrices in the noisy case. The procedure is adaptive to the rank and robust against small lowrank perturbations. Both upper and lower bounds for the estimation accuracy under the Frobenius norm loss are obtained. The proposed estimator is shown to be rateoptimal under certain conditions. The estimator is easy to implement via convex programming and performs well numerically. The main results obtained in the paper also have implications to other related statistical problems. An application to estimation of spike covariance matrices from onedimensional random projections is considered. The results demonstrate that it is possible to accurately estimate the covariance matrix of a highdimensional distribution based only on onedimensional projections.
Improved Analyses for SP and CoSaMP Algorithms in Terms of Restricted Isometry Constants
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Compressed Sensing Recovery via Nonconvex Shrinkage Penalties
"... The `0 minimization of compressed sensing is often relaxed to `1, which yields easy computation using the shrinkage mapping known as soft thresholding, and can be shown to recover the original solution under certain hypotheses. Recent work has derived a general class of shrinkages and associated non ..."
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The `0 minimization of compressed sensing is often relaxed to `1, which yields easy computation using the shrinkage mapping known as soft thresholding, and can be shown to recover the original solution under certain hypotheses. Recent work has derived a general class of shrinkages and associated nonconvex penalties that better approximate the original `0 penalty and empirically can recover the original solution from fewer measurements. We specifically examine pshrinkage and firm thresholding. In this work, we prove that given data and a measurement matrix from a broad class of matrices, one can choose parameters for these classes of shrinkages to guarantee exact recovery of the sparsest solution. We further prove convergence of the algorithm iterative pshrinkage (IPS) for solving one such relaxed problem.
Sparse Recovery via Partial Regularization: Models, Theory and Algorithms ∗
, 2015
"... In the context of sparse recovery, it is known that most of existing regularizers such as `1 suffer from some bias incurred by some leading entries (in magnitude) of the associated vector. To neutralize this bias, we propose a class of models with partial regularizers for recovering a sparse solutio ..."
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In the context of sparse recovery, it is known that most of existing regularizers such as `1 suffer from some bias incurred by some leading entries (in magnitude) of the associated vector. To neutralize this bias, we propose a class of models with partial regularizers for recovering a sparse solution of a linear system. We show that every local minimizer of these models is sufficiently sparse or the magnitude of all its nonzero entries is above a uniform constant depending only on the data of the linear system. Moreover, for a class of partial regularizers, any global minimizer of these models is a sparsest solution to the linear system. We also establish some sufficient conditions for local or global recovery of the sparsest solution to the linear system, among which one of the conditions is weaker than the best known restricted isometry property (RIP) condition for sparse recovery by `1. In addition, a firstorder feasible augmented Lagrangian (FAL) method is proposed for solving these models, in which each subproblem is solved by a nonmonotone proximal gradient (NPG) method. Despite the complication of the partial regularizers, we show that each proximal subproblem in NPG can be solved as a certain number of onedimensional optimization problems, which usually have a closedform solution. We also show that any accumulation point of the sequence generated by FAL is a firstorder stationary point of the models. Numerical results on compressed sensing and sparse logistic regression demonstrate that the proposed models substantially outperform the widely used ones in the literature in terms of solution quality.
Surface Reconstruction in GradientField Domain Using Compressed Sensing
"... Abstract — Surface reconstruction from measurements of spatial gradient is an important computer vision problem with applications in photometric stereo and shapefromshading. In the case of morphologically complex surfaces observed in the presence of shadowing and transparency artifacts, a relativ ..."
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Abstract — Surface reconstruction from measurements of spatial gradient is an important computer vision problem with applications in photometric stereo and shapefromshading. In the case of morphologically complex surfaces observed in the presence of shadowing and transparency artifacts, a relatively large dense gradient measurements may be required for accurate surface reconstruction. Consequently, due to hardware limitations of image acquisition devices, situations are possible in which the available sampling density might not be sufficiently high to allow for recovery of essential surface details. In this paper, the above problem is resolved by means of derivative compressed sensing (DCS). DCS can be viewed as a modification of the classical CS, which is particularly suited for reconstructions involving image/surface gradients. In DCS, a standard CS setting is augmented through incorporation of additional constraints arising from some intrinsic properties of potential vector fields. We demonstrate that using DCS results in reduction in the number of measurements as compared with the standard (dense) sampling, while producing estimates of higher accuracy and smaller variability as compared with CSbased estimates. The results of this study are further supported by a series of numerical experiments. Index Terms — Photometric stereo, shapefromshading, 3D surface reconstruction, derivative compressed sensing,
A L1Minimization for Image Reconstruction from Compressed Sensing
, 2013
"... Abstract: We proposed a simple and efficient iteratively reweighted algorithm to improve the recovery performance for image reconstruction from compressive sensing (CS). The numerical experiential results demonstrate that the new proposed method outperforms in image quality and computation complexit ..."
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Abstract: We proposed a simple and efficient iteratively reweighted algorithm to improve the recovery performance for image reconstruction from compressive sensing (CS). The numerical experiential results demonstrate that the new proposed method outperforms in image quality and computation complexity, compared with standard l1minimization and other iteratively reweighted l1algorithms when applying for image