Sparse Signal Estimation by Maximally Sparse Convex Optimization

Selesnick, Ivan; Bayram, İlker

doi:10.1109/tsp.2014.2298839

Cited by 162 publications

(132 citation statements)

References 65 publications

(108 reference statements)

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“…The Logarithmic and Arctangent (parametric) penalty functions were introduced in [7]; it also focused on the conditions to be met by non-convex penalty functions so as to ensure the convexity of the total cost function of (1). While these penalty functions exhibit less bias (than the 1 -norm) they are particularly convenient in algorithms for solving (1) that do not used their corresponding thresholding rules directly but the derivative of their penalty functions, such IRLS [14], FOCUSS [15] and (majorization-minimization) MM-based [16].…”

Section: A Non 1 -Norm Penalty Functionsmentioning

confidence: 99%

“…are the same as for (1). Recently, several works [5], [6], [7], [8] have proposed or assessed the use of different penalty functions for (1); the list includes the Logarithmic and Arctangent penalty functions [6], as well as those associated with the Non-negative Garrote (NNG) [9], SCAD [10] and Firm [11] thresholding rules. The key sought-after property for these alternatives is to induce sparsity more strongly than the 1 -norm penalty function.…”

Section: Introductionmentioning

confidence: 99%

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A two-term penalty function for inverse problems with sparsity constrains

Rodríguez

2017

2017 25th European Signal Processing Conference (EUSIPCO)

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Abstract-Inverse problems with sparsity constrains, such Basis Pursuit denoising (BPDN) and Convolutional BPDN (CBPDN), usually use the 1-norm as the penalty function; however such choice leads to a solution that is biased towards zero. Recently, several works have proposed and assessed the properties of other non-standard penalty functions (most of them non-convex), which avoid the above mentioned drawback and at the same time are intended to induce sparsity more strongly than the 1-norm.In this paper we propose a two-term penalty function consisting of a synthesis between the 1-norm and the penalty function associated with the Non-Negative Garrote (NNG) thresholding rule. Although the proposed two-term penalty function is nonconvex, the total cost function for the BPDN / CBPDN problems is still convex. The performance of the proposed twoterm penalty function is compared with other reported choices for practical denoising, deconvolution and convolutional sparse coding (CSC) problems within the BPDN / CBPDN frameworks. Our experimental results show that the proposed two-term penalty function is particularly effective (better reconstruction with sparser solutions) for the CSC problem while attaining competitive performance for the denoising and deconvolution problems.

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Section: A Non 1 -Norm Penalty Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A two-term penalty function for inverse problems with sparsity constrains

Rodríguez

2017

2017 25th European Signal Processing Conference (EUSIPCO)

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“…As , the log and atan penalties approach the absolute value function. The atan penalty was derived to promote sparsity more strongly than the log penalty [33]. It will be useful below to define as .…”

Section: A Problem Formulationmentioning

confidence: 99%

“…Hence, non-convex penalties should be specified with care. One approach to avoid the issue of entrapment in local minima is to specify non-convex penalties such that the total objective function, , is convex [7], [26], [27], [33]. Then the total objective function, owing to its convexity, does not posses sub-optimal local minima and a global optimal solution can be reliably found.…”

Section: E Setting the Non-convexity Parametersmentioning

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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“…As a result, these methods suffer from numerical problems for smaller values of p. Thus, an attractive solution is to employ Lipschitz continuous approximations, such as the exponential function, the logarithm function or sigmoid functions, e.g., [20,23,34]. The arctan function is also used in different literature works for sparse regularization, such as approximating the sign function appearing in the derivative of the 1 -norm term in [35], introducing a penalty function for the sparse signal estimation by the maximally-sparse convex approach in [36] or approximating the 0 -norm term through a weighted 1 -norm term in [23].…”

Section: Introductionmentioning

confidence: 99%

ℓ0-Norm Sparse Hyperspectral Unmixing Using Arctan Smoothing

et al. 2016

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Abstract:The goal of sparse linear hyperspectral unmixing is to determine a scanty subset of spectral signatures of materials contained in each mixed pixel and to estimate their fractional abundances. This turns into an 0 -norm minimization, which is an NP-hard problem. In this paper, we propose a new iterative method, which starts as an 1 -norm optimization that is convex, has a unique solution, converges quickly and iteratively tends to be an 0 -norm problem. More specifically, we employ the arctan function with the parameter σ ≥ 0 in our optimization. This function is Lipschitz continuous and approximates 1 -norm and 0 -norm for small and large values of σ, respectively. We prove that the set of local optima of our problem is continuous versus σ. Thus, by a gradual increase of σ in each iteration, we may avoid being trapped in a suboptimal solution. We propose to use the alternating direction method of multipliers (ADMM) for our minimization problem iteratively while increasing σ exponentially. Our evaluations reveal the superiorities and shortcomings of the proposed method compared to several state-of-the-art methods. We consider such evaluations in different experiments over both synthetic and real hyperspectral data, and the results of our proposed methods reveal the sparsest estimated abundances compared to other competitive algorithms for the subimage of AVIRIS cuprite data.

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Sparse Signal Estimation by Maximally Sparse Convex Optimization

Cited by 162 publications

References 65 publications

A two-term penalty function for inverse problems with sparsity constrains

A two-term penalty function for inverse problems with sparsity constrains

Transient Artifact Reduction Algorithm (TARA) Based on Sparse Optimization

ℓ0-Norm Sparse Hyperspectral Unmixing Using Arctan Smoothing

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