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34
Guaranteed minimumrank solutions of linear matrix equations via nuclear norm minimization
, 2007
"... The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative ..."
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Cited by 218 (15 self)
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The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative filtering. Although specific instances can often be solved with specialized algorithms, the general affine rank minimization problem is NPhard, because it contains vector cardinality minimization as a special case. In this paper, we show that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum rank solution can be recovered by solving a convex optimization problem, namely the minimization of the nuclear norm over the given affine space. We present several random ensembles of equations where the restricted isometry property holds with overwhelming probability, provided the codimension of the subspace is sufficiently large. The techniques used in our analysis have strong parallels in the compressed sensing framework. We discuss how affine rank minimization generalizes this preexisting concept and outline a dictionary relating concepts from cardinality minimization to those of rank minimization. We also discuss several algorithmic approaches to solving the norm minimization relaxations, and illustrate our results with numerical examples.
Iteratively reweighted algorithms for compressive sensing
 in 33rd International Conference on Acoustics, Speech, and Signal Processing (ICASSP
, 2008
"... The theory of compressive sensing has shown that sparse signals can be reconstructed exactly from many fewer measurements than traditionally believed necessary. In [1], it was shown empirically that using ℓ p minimization with p < 1 can do so with fewer measurements than with p = 1. In this paper we ..."
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Cited by 75 (6 self)
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The theory of compressive sensing has shown that sparse signals can be reconstructed exactly from many fewer measurements than traditionally believed necessary. In [1], it was shown empirically that using ℓ p minimization with p < 1 can do so with fewer measurements than with p = 1. In this paper we consider the use of iteratively reweighted algorithms for computing local minima of the nonconvex problem. In particular, a particular regularization strategy is found to greatly improve the ability of a reweighted leastsquares algorithm to recover sparse signals, with exact recovery being observed for signals that are much less sparse than required by an unregularized version (such as FOCUSS, [2]). Improvements are also observed for the reweightedℓ 1 approach of [3]. Index Terms — Compressive sensing, signal reconstruction, nonconvex optimization, iteratively reweighted least squares, ℓ 1 minimization. 1.
ℓ1 Trend Filtering
, 2007
"... The problem of estimating underlying trends in time series data arises in a variety of disciplines. In this paper we propose a variation on HodrickPrescott (HP) filtering, a widely used method for trend estimation. The proposed ℓ1 trend filtering method substitutes a sum of absolute values (i.e., ..."
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Cited by 18 (6 self)
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The problem of estimating underlying trends in time series data arises in a variety of disciplines. In this paper we propose a variation on HodrickPrescott (HP) filtering, a widely used method for trend estimation. The proposed ℓ1 trend filtering method substitutes a sum of absolute values (i.e., an ℓ1norm) for the sum of squares used in HP filtering to penalize variations in the estimated trend. The ℓ1 trend filtering method produces trend estimates that are piecewise linear, and therefore is well suited to analyzing time series with an underlying piecewise linear trend. The kinks, knots, or changes in slope, of the estimated trend can be interpreted as abrupt changes or events in the underlying dynamics of the time series. Using specialized interiorpoint methods, ℓ1 trend filtering can be carried out with not much more effort than HP filtering; in particular, the number of arithmetic operations required grows linearly with the number of data points. We describe the method and some of its basic properties, and give some illustrative examples. We show how the method is related to ℓ1 regularization based methods in sparse signal recovery and feature selection, and list some extensions of the basic method.
Learning with Compressible Priors
"... We describe a set of probability distributions, dubbed compressible priors, whose independent and identically distributed (iid) realizations result in pcompressible signals. A signal x ∈ R N is called pcompressible with magnitude R if its sorted coefficients exhibit a powerlaw decay as x(i) � ..."
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Cited by 15 (4 self)
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We describe a set of probability distributions, dubbed compressible priors, whose independent and identically distributed (iid) realizations result in pcompressible signals. A signal x ∈ R N is called pcompressible with magnitude R if its sorted coefficients exhibit a powerlaw decay as x(i) � R · i −d, where the decay rate d is equal to 1/p. pcompressible signals live close to Ksparse signals (K ≪ N) in the ℓrnorm (r> p) since their best Ksparse approximation error decreases with O ( R · K 1/r−1/p). We show that the membership of generalized Pareto, Student’s t, lognormal, Fréchet, and loglogistic distributions to the set of compressible priors depends only on the distribution parameters and is independent of N. In contrast, we demonstrate that the membership of the generalized Gaussian distribution (GGD) depends both on the signal dimension and the GGD parameters: the expected decay rate of Nsample iid realizations from the GGD with the shape parameter q is given by 1 / [q log (N/q)]. As stylized examples, we show via experiments that the wavelet coefficients of natural images are 1.67compressible whereas their pixel gradients are 0.95 log (N/0.95)compressible, on the average. We also leverage the connections between compressible priors and sparse signals to develop new iterative reweighted sparse signal recovery algorithms that outperform the standard ℓ1norm minimization. Finally, we describe how to learn the hyperparameters of compressible priors in underdetermined regression problems by exploiting the geometry of their order statistics during signal recovery. 1
Reducing the risk of query expansion via robust constrained optimization
 Proceedings of the Eighteenth International Conference on Information and Knowledge Management (CIKM 2009). ACM. Hong
"... We introduce a new theoretical derivation, evaluation methods, and extensive empirical analysis for an automatic query expansion framework in which model estimation is cast as a robust constrained optimization problem. This framework provides a powerful method for modeling and solving complex expans ..."
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Cited by 13 (5 self)
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We introduce a new theoretical derivation, evaluation methods, and extensive empirical analysis for an automatic query expansion framework in which model estimation is cast as a robust constrained optimization problem. This framework provides a powerful method for modeling and solving complex expansion problems, by allowing multiple sources of domain knowledge or evidence to be encoded as simultaneous optimization constraints. Our robust optimization approach provides a clean theoretical way to model not only expansion benefit, but also expansion risk, by optimizing over uncertainty sets for the data. In addition, we introduce riskreward curves to visualize expansion algorithm performance and analyze parameter sensitivity. We show that a robust approach significantly reduces the number and magnitude of expansion failures for a strong baseline algorithm, with no loss in average gain. Our approach is implemented as a highly efficient postprocessing step that assumes little about the baseline expansion method used as input, making it easy to apply to existing expansion methods. We provide analysis showing that this approach is a natural and effective way to do selective expansion, automatically reducing or avoiding expansion in risky scenarios, and successfully attenuating noise in poor baseline methods.
Face recognition with contiguous occlusion using markov random fields
 in Proceedings of IEEE International Conference on Computer Vision, 2009
"... Partially occluded faces are common in many applications of face recognition. While algorithms based on sparse representation have demonstrated promising results, they achieve their best performance on occlusions that are not spatially correlated (i.e. random pixel corruption). We show that such spa ..."
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Cited by 9 (3 self)
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Partially occluded faces are common in many applications of face recognition. While algorithms based on sparse representation have demonstrated promising results, they achieve their best performance on occlusions that are not spatially correlated (i.e. random pixel corruption). We show that such sparsitybased algorithms can be significantly improved by harnessing prior knowledge about the pixel error distribution. We show how a Markov Random Field model for spatial continuity of the occlusion can be integrated into the computation of a sparse representation of the test image with respect to the training images. Our algorithm efficiently and reliably identifies the corrupted regions and excludes them from the sparse representation. Extensive experiments on both laboratory and realworld datasets show that our algorithm tolerates much larger fractions and varieties of occlusion than current stateoftheart algorithms. 1.
CurveletWavelet Regularized Split Bregman Iteration for Compressed Sensing
"... Compressed sensing is a new concept in signal processing. Assuming that a signal can be represented or approximated by only a few suitably chosen terms in a frame expansion, compressed sensing allows to recover this signal from much fewer samples than the ShannonNyquist theory requires. Many images ..."
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Cited by 8 (3 self)
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Compressed sensing is a new concept in signal processing. Assuming that a signal can be represented or approximated by only a few suitably chosen terms in a frame expansion, compressed sensing allows to recover this signal from much fewer samples than the ShannonNyquist theory requires. Many images can be sparsely approximated in expansions of suitable frames as wavelets, curvelets, wave atoms and others. Generally, wavelets represent pointlike features while curvelets represent linelike features well. For a suitable recovery of images, we propose models that contain weighted sparsity constraints in two different frames. Given the incomplete measurements f = Φu + ɛ with the measurement matrix Φ ∈ R K×N, K<<N, we consider a jointly sparsityconstrained optimization problem of the form argmin{‖ΛcΨcu‖1 + ‖ΛwΨwu‖1 + u 1 2‖f − Φu‖22}. Here Ψcand Ψw are the transform matrices corresponding to the two frames, and the diagonal matrices Λc, Λw contain the weights for the frame coefficients. We present efficient iteration methods to solve the optimization problem, based on Alternating Split Bregman algorithms. The convergence of the proposed iteration schemes will be proved by showing that they can be understood as special cases of the DouglasRachford Split algorithm. Numerical experiments for compressed sensing based Fourierdomain random imaging show good performances of the proposed curveletwavelet regularized split Bregman (CWSpB) methods,whereweparticularlyuseacombination of wavelet and curvelet coefficients as sparsity constraints.
Recovering sparse signals with a certain family of nonconvex penalties and DC programming
 IEEE TRANSACTIONS ON SIGNAL PROCESSING
"... This paper considers the problem of recovering a sparse signal representation according to a signal dictionary. This problem could be formalized as a penalized leastsquares problem in which sparsity is usually induced by a ℓ1norm penalty on the coefficients. Such an approach known as the Lasso or ..."
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Cited by 8 (2 self)
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This paper considers the problem of recovering a sparse signal representation according to a signal dictionary. This problem could be formalized as a penalized leastsquares problem in which sparsity is usually induced by a ℓ1norm penalty on the coefficients. Such an approach known as the Lasso or Basis Pursuit Denoising has been shown to perform reasonably well in some situations. However, it was also proved that nonconvex penalties like the pseudo ℓqnorm with q < 1 or SCAD penalty are able to recover sparsity in a more efficient way than the Lasso. Several algorithms have been proposed for solving the resulting nonconvex leastsquares problem. This paper proposes a generic algorithm to address such a sparsity recovery problem for some class of nonconvex penalties. Our main contribution is that the proposed methodology is based on an iterative algorithm which solves at each iteration a convex weighted Lasso problem. It relies on the family of nonconvex penalties which can be decomposed as a difference of convex functions. This allows us to apply difference of convex functions programming which is a generic and principled way for solving nonsmooth and nonconvex optimization problem. We also show that several algorithms in the literature dealing with nonconvex penalties are particular instances of our algorithm. Experimental results demonstrate the effectiveness of the proposed generic framework compared to existing algorithms, including iterative reweighted leastsquares methods.