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 NP-hard, 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 pre-existing 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.

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Seismic data recovery from data with missing traces on otherwise regular acquisition grids forms a crucial step in the seismic processing flow. For instance, unsuccesful recovery leads to imaging artifacts and to erroneous predictions for the multiples, adversely affecting the performance of multiple ellimination. A non-parametric transform-based recovery method is presented that exploits the compression of seismic data volumes by multidimensional expansions with respect to recently developed curvelet frames. The frame elements of these transforms locally resemble wavefronts present in the data and this leads to a compressible signal representation. This compression enables us to formulate a new seismic data recovery algorithm through sparsity-promoting inversion. The concept of sparsity-promoting inversion is in itself not new to the geosciences. However, the recent insights from the field of ‘compressed sensing ’ are new since they identify the conditions that determine successful recovery. These conditions are carefully examined by means of examples geared towards the seismic recovery problem for data with large percentages (>70 %) of traces missing. We show that as long as there is sufficient ’randomness ’ in the acquistion pattern, recovery to within an acceptable error is possible. We also show that our approach compares favor-

It is now well understood that (1) it is possible to reconstruct sparse signals exactly from what appear to be highly incomplete sets of linear measurements and (2) that this can be done by constrained ℓ1 minimization. In this paper, we study a novel method for sparse signal recovery that in many situations outperforms ℓ1 minimization in the sense that substantially fewer measurements are needed for exact recovery. The algorithm consists of solving a sequence of weighted ℓ1-minimization problems where the weights used for the next iteration are computed from the value of the current solution. We present a series of experiments demonstrating the remarkable performance and broad applicability of this algorithm in the areas of sparse signal recovery, statistical estimation, error correction and image processing. Interestingly, superior gains are also achieved when our method is applied to recover signals with assumed near-sparsity in overcomplete representations—not by reweighting the ℓ1 norm of the coefficient sequence as is common, but by reweighting the ℓ1 norm of the transformed object. An immediate consequence is the possibility of highly efficient data acquisition protocols by improving on a technique known as compressed sensing.

We consider the problem of estimating a sparse signal from a set of quantized, Gaussian noise corrupted measurements, where each measurement corresponds to an interval of values. We give two methods for (approximately) solving this problem, each based on minimizing a differentiable convex function plus an regularization term. Using a first order method developed by Hale et al, we demonstrate the performance of the methods through numerical simulation. We find that, using these methods, compressed sensing can be carried out even when the quantization is very coarse, e.g., 1 or 2 bits per measurement.

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A historical account is given of the development of methods for solving approximation problems set in normed linear spaces. Approximation of both real functions and real data is considered, with particular reference to L p (or l p ) and Chebyshev norms. As well as coverage of methods for the usual linear problems, an account is given of the development of methods for approximation by functions which are nonlinear in the free parameters, and special attention is paid to some particular nonlinear approximating families. 1 Introduction The purpose of this paper is to give a historical account of the development of numerical methods for a range of problems in best approximation, that is problems which involve the minimization of a norm. A treatment is given of approximation of both real functions and data. For the approximation of functions, the emphasis is on the use of the Chebyshev norm, while for data approximation, we consider a wider range of criteria, including the other l ...

Running mean filters have long been used as time domain, digital, low-pass filters. They are a convolution in the time domain with a boxcar function (or, more generally, with functions designed for better frequency characteristics). Running means are linear and easily described by their transfer functions in the frequency domain. They handle some kinds of high-frequency contamination well. However, faced with the spikey noise sometimes affecting electronic systems and phenomena subject to interference by lightening, cosmic rays, and timing tick marks running means usually perform poorly: spikes become low, wide, one-sided pulses affecting a broad region of the signal in the neighborhood of an originally short glitch. Since we can be generally confident of the signal except during the spike itself, a better triter would be one that does not affect the signal outside the spike, but replaces only the bad data with some interpolation of the good data. Stated another way, this filter would do what the human data editor does. Running medians nearly behave like this ideal filter; a simple extension of them is an ideal, general despiker. Claerbout and Muir (1973) discussed the use of medians and the related L1 norm