Abstract. The time-frequency and time-scale communities have recently developed a large number of overcomplete waveform dictionaries — stationary wavelets, wavelet packets, cosine packets, chirplets, and warplets, to name a few. Decomposition into overcomplete systems is not unique, and several methods for decomposition have been proposed, including the method of frames (MOF), Matching pursuit (MP), and, for special dictionaries, the best orthogonal basis (BOB). Basis Pursuit (BP) is a principle for decomposing a signal into an “optimal ” superposition of dictionary elements, where optimal means having the smallest l 1 norm of coefficients among all such decompositions. We give examples exhibiting several advantages over MOF, MP, and BOB, including better sparsity and superresolution. BP has interesting relations to ideas in areas as diverse as ill-posed problems, in abstract harmonic analysis, total variation denoising, and multiscale edge denoising. BP in highly overcomplete dictionaries leads to large-scale optimization problems. With signals of length 8192 and a wavelet packet dictionary, one gets an equivalent linear program of size 8192 by 212,992. Such problems can be attacked successfully only because of recent advances in linear programming by interior-point methods. We obtain reasonable success with a primal-dual logarithmic barrier method and conjugate-gradient solver.

This paper considers the classical error correcting problem which is frequently discussed in coding theory. We wish to recover an input vector f ∈ Rn from corrupted measurements y = Af + e. Here, A is an m by n (coding) matrix and e is an arbitrary and unknown vector of errors. Is it possible to recover f exactly from the data y? We prove that under suitable conditions on the coding matrix A, the input f is the unique solution to the ℓ1-minimization problem (‖x‖ℓ1:= i |xi|) min g∈R n ‖y − Ag‖ℓ1 provided that the support of the vector of errors is not too large, ‖e‖ℓ0: = |{i: ei ̸= 0} | ≤ ρ · m for some ρ> 0. In short, f can be recovered exactly by solving a simple convex optimization problem (which one can recast as a linear program). In addition, numerical experiments suggest that this recovery procedure works unreasonably well; f is recovered exactly even in situations where a significant fraction of the output is corrupted. This work is related to the problem of finding sparse solutions to vastly underdetermined systems of linear equations. There are also significant connections with the problem of recovering signals from highly incomplete measurements. In fact, the results introduced in this paper improve on our earlier work [5]. Finally, underlying the success of ℓ1 is a crucial property we call the uniform uncertainty principle that we shall describe in detail.

The Time-Frequency and Time-Scale communities have recently developed an enormous number of overcomplete signal dictionaries -- wavelets, wavelet packets, cosine packets, wilson bases, chirplets, warped bases, and hyperbolic cross bases being a few examples. Basis Pursuit is a technique for decomposing a signal into an "optimal" superposition of dictionary elements. The optimization criterion is the l 1 norm of coefficients. The method has several advantages over Matching Pursuit and Best Ortho Basis, including super-resolution and stability. 1 Introduction Over the last five years or so, there has been an explosion of awareness of alternatives to traditional signal representations. Instead of just representing objects as superpositions of sinusoids (the traditional Fourier representation) we now have available alternate dictionaries -- signal representation schemes -- of which the Wavelets dictionary is only the most well-known. Wavelet dictionaries, Gabor dictionaries, Multi-scale...

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We review recent methods for learning with positive definite kernels. All these methods formulate learning and estimation problems as linear tasks in a reproducing kernel Hilbert space (RKHS) associated with a kernel. We cover a wide range of methods, ranging from simple classifiers to sophisticated methods for estimation with structured data.

We introduce some recent and very recent smoothing methods which focus on the preservation of boundaries, spikes and canyons in presence of noise. We try to point out basic principles they have in common; the most important one is the robustness aspect. It is reflected by the use of `cup functions' in the statistical loss functions instead of squares; such cup functions were introduced early in robust statistics to downweight outliers. Basically, they are variants of truncated squares. We discuss all the methods in the common framework of `energy functions', i.e we associate to (most of ) the algorithms a `loss function' in such a fashion that the output of the algorithm or the `estimate' is a global or local minimum of this loss function. The third aspect we pursue is the correspondence between loss functions and their local minima and nonlinear filters. We shall argue that the nonlinear filters can be interpreted as variants of gradient descent on the loss functions. This way we can ...

performance modeling, response time, distributed applications, capacity management Distributed applications play an increasingly crucial role in businesscritical enterprise operations. Understanding the performance of such applications is challenging yet essential due to their growing economic value. A particularly important problem is performance prediction: How will application-level performance vary in response to changes in workload? This paper presents a practical and accurate approach to predicting response times as a function of workload mix in complex modern distributed applications. We compare our approach against several alternatives, evaluating their ability to predict the performance of two large, real business-critical production systems and a testbed application subjected to extremely variable synthetic workload. Our results show that our method yields accurate response time predictions under a wide range of conditions, and that our models generalize well to previously-unseen regions of workload/performance space.

participants at Brookings, Northwestern, TAPES, and Tax Economists ’ Forum. We also thank Tats Kanenari and Norma Coe for outstanding research assistance, Bill Even for providing the CPS data extract used in the paper, and Stacy Furukawa for assistance with the data. Gale gratefully acknowledges financial support from the National Institute on Aging under grant AG11836. All opinions are our own and should not be attributed to the staff, officers, or trustees of the Federal Reserve Board or the This paper provides a new econometric specification and new evidence on the impact of 401(k) plans on household wealth. We allow the impact of 401(k)s to vary over both time and earnings groups. Our specification--motivated by a variety of theoretical considerations and data patterns--generalizes earlier work in the literature, and we show that the modeling constraints imposed by previous authors are rejected by the data. Using data from 1987 and 1991 from the Survey of Income and Program Participation, we find that the effects of 401(k)s on household wealth vary significantly by earnings level. Our analysis implies that 401(k)s held by groups with low earnings, who hold a small portion of 401(k) balances, are more likely to represent additions to net wealth than 401(k)s held by high-earning groups, who hold the bulk of 401(k) assets. Thus, between 0 and 30 percent of 401(k) balances represent net additions to private

In this article we consider quantile regression in reproducing kernel Hilbert spaces, which we call kernel quantile regression (KQR). We make three contributions: (1) we propose an efficient algorithm that computes the entire solution path of the KQR, with essentially the same computational cost as fitting one KQR model; (2) we derive a simple formula for the effective dimension of the KQR model, which allows convenient selection of the regularization parameter; and (3) we develop an asymptotic theory for the KQR model.