Convergence of a Sparse Representations Algorithm Applicable to Real or Complex Data

This paper develops a convex approach for sparse one-dimensional deconvolution that improves upon L1-norm regularization, the standard convex approach. We propose a sparsity-inducing non-separable non-convex bivariate penalty function for this purpose. It is designed to enable the convex formulation of ill-conditioned linear inverse problems with quadratic data fidelity terms. The new penalty overcomes limitations of separable regularization. We show how the penalty parameters should be set to ensure that the objective function is convex, and provide an explicit condition to verify the optimality of a prospective solution. We present an algorithm (an instance of forward-backward splitting) for sparse deconvolution using the new penalty.

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“…which follows from an analysis of sparse optimality conditions [36,37,70]. We set β = 2.5, similar to the 'three sigma' rule.…”

Section: Examplementioning

confidence: 99%

Enhanced Sparsity by Non-Separable Regularization

Selesnick

Bayram

2016

IEEE Trans. Signal Process.

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“…We illustrate this possibility by considering a classical signal processing problem where the goal is to estimate the number, amplitude, and initial phase of a set of superimposed sinusoids, observed under noise [25], [41]. A discrete formulation of this problem may be given the form (2), where matrix A is complex, of size k × 2m f (where m f is the maximum frequency), with elements given by…”

Section: Problems With Complex Datamentioning

confidence: 99%

Sparse Reconstruction by Separable Approximation

Wright

Nowak

Figueiredo

2009

IEEE Trans. Signal Process.

1,459

307

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Abstract-Finding sparse approximate solutions to large underdetermined linear systems of equations is a common problem in signal/image processing and statistics. Basis pursuit, the least absolute shrinkage and selection operator (LASSO), waveletbased deconvolution and reconstruction, and compressed sensing (CS) are a few well-known areas in which problems of this type appear. One standard approach is to minimize an objective function that includes a quadratic ( 2) error term added to a sparsity-inducing (usually 1) regularization term. We present an algorithmic framework for the more general problem of minimizing the sum of a smooth convex function and a nonsmooth, possibly nonconvex regularizer. We propose iterative methods in which each step is obtained by solving an optimization subproblem involving a quadratic term with diagonal Hessian (which is therefore separable in the unknowns) plus the original sparsity-inducing regularizer. Our approach is suitable for cases in which this subproblem can be solved much more rapidly than the original problem. In addition to solving the standard 2 − 1 case, our framework yields an efficient solution technique for other regularizers, such as an ∞-norm regularizer and groupseparable (GS) regularizers. It also generalizes immediately to the case in which the data is complex rather than real. Experiments with CS problems show that our approach is competitive with the fastest known methods for the standard 2 − 1 problem, as well as being efficient on problems with other separable regularization terms.

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“…Hence, fast solvers for banded systems cannot be used directly with (18). To utilize fast solvers, we write (22) where the matrix within the inverse is banded.…”

Section: B Algorithmmentioning

confidence: 99%

Transient Artifact Reduction Algorithm (TARA) Based on Sparse Optimization

Selesnick¹,

Graber

Ding³

et al. 2014

IEEE Trans. Signal Process.

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Abstract-This paper addresses the suppression of transient artifacts in signals, e.g., biomedical time series. To that end, we distinguish two types of artifact signals. We define "Type 1" artifacts as spikes and sharp, brief waves that adhere to a baseline value of zero. We define "Type 2" artifacts as comprising approximate step discontinuities. We model a Type 1 artifact as being sparse and having a sparse time-derivative, and a Type 2 artifact as having a sparse time-derivative. We model the observed time series as the sum of a low-pass signal (e.g., a background trend), an artifact signal of each type, and a white Gaussian stochastic process. To jointly estimate the components of the signal model, we formulate a sparse optimization problem and develop a rapidly converging, computationally efficient iterative algorithm denoted TARA ("transient artifact reduction algorithm"). The effectiveness of the approach is illustrated using near infrared spectroscopic time-series data.

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Convergence of a Sparse Representations Algorithm Applicable to Real or Complex Data

Cited by 31 publications

References 39 publications

Enhanced Sparsity by Non-Separable Regularization

Enhanced Sparsity by Non-Separable Regularization

Sparse Reconstruction by Separable Approximation

Transient Artifact Reduction Algorithm (TARA) Based on Sparse Optimization

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