Properties and Iterative Methods for the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>Q</mml:mi></mml:mrow></mml:math>-Lasso

Alghamdi, Maryam A.; Alghamdi, Mohammad A.; Shahzad, Naseer; Xu, Hong–Kun

doi:10.1155/2013/250943

Cited by 5 publications

(10 citation statements)

References 13 publications

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“…In this paper we investigate split feasibility problems under a nonconvex Lipschitz continuous metric instead of conventional methods such as l 1 or l 1 − l 2 minimization, for example in [1]. We present and analyze the convergence to a stationary point of an iterative minimization method based on DCA (difference of convex algorithm), see for example [20]).…”

Section: Discussionmentioning

confidence: 99%

“…where ε > 0 is the tolerance level of errors and p is often 1, 2 or ∞. It is noticed in [1] that if we let Q := B ε (b), the closed ball in IR n with center b and radius ε, then (1.5) is rewritten as…”

Section: Introductionmentioning

confidence: 99%

“…with γ > 0 a Lagrangian multiplier. A connection is also made in [1] with the so-called split feasibility problem [5] which is stated as finding x verifying…”

Section: Introductionmentioning

confidence: 99%

“…where γ > 0 is a regularization parameter. This convex relaxation attracts considerable attention see for example [1] and references there in. In this paper we study a non-convex but Lipschitz continuous metric l 1 -l 2 for SFP.…”

Section: Introductionmentioning

confidence: 99%

“…In [1], properties and iterative methods for (1.7) are investigated. Remember also that many authors devoted their works to the unconstrained minimization problem min x∈H f 1 (x) + f 2 (x) with f 1 , f 2 are two proper, convex lower semi continuous functions defined on a Hilbert space H and f 2 differentiable with a β-Lipschitz continuous gradient for some β > 0 and an effective method to solve it is the forward-backward algorithm which from an initial value x 0 generates a sequence (x k ) by the following iteration…”

Section: Introductionmentioning

confidence: 99%

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l 1-l 2 regularization of split feasibility problems

Moudafi

Gibali

2017

Numer Algor

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Numerous problems in signal processing and imaging, statistical learning and data mining, or computer vision can be formulated as optimization problems which consist in minimizing a sum of convex functions, not necessarily differentiable, possibly composed with linear operators and that in turn can be transformed to split feasibility problems (SFP), see for example [5]. Each function is typically either a data fidelity term or a regularization term enforcing some properties on the solution, see for example [9] and references therein. In this paper we are interested in Split Feasibility Problems which can be seen as a general form of Q-Lasso introduced in [1] that extended the well-known Lasso of Tibshirani [24]. Q is a closed convex subset of a Euclidean m-space, for some integer m ≥ 1, that can be interpreted as the set of errors within given tolerance level when linear measurements are taken to recover a signal/image via the Lasso. Inspired by recent works by Lou et al [16,26], we are interested in a nonconvex regularization of SFP and propose three split algorithms for solving this general case. The first one is based on the DC (difference of convex) algorithm (DCA) introduced by Pham Dinh Tao, the second one in nothing else than the celebrate forward-backward algorithm and the third one uses a method introduced by Mine and Fukushima. It is worth mentioning that the SFP model a number of applied problems arising from signal/image processing and specially optimization problems for intensity-modulated radiation therapy (IMRT) treatment planning, see for example [4].

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…with γ > 0 a Lagrangian multiplier. A connection is also made in [1] with the so-called split feasibility problem [5] which is stated as finding x verifying…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

l 1-l 2 regularization of split feasibility problems

Moudafi

Gibali

2017

Numer Algor

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Difference of two norms-regularizations forQ-Lasso

Moudafi

2020

ACI

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The focus of this paper is in Q-Lasso introduced in Alghamdi et al. (2013) which extended the Lasso by Tibshirani (1996). The closed convex subset Q belonging in a Euclidean m-space, for m∈IN, is the set of errors when linear measurements are taken to recover a signal/image via the Lasso. Based on a recent work by Wang (2013), we are interested in two new penalty methods for Q-Lasso relying on two types of difference of convex functions (DC for short) programming where the DC objective functions are the difference of l1 and lσq norms and the difference of l1 and lr norms with r>1. By means of a generalized q-term shrinkage operator upon the special structure of lσq norm, we design a proximal gradient algorithm for handling the DC l1−lσq model. Then, based on the majorization scheme, we develop a majorized penalty algorithm for the DC l1−lr model. The convergence results of our new algorithms are presented as well. We would like to emphasize that extensive simulation results in the case Q={b} show that these two new algorithms offer improved signal recovery performance and require reduced computational effort relative to state-of-the-art l1 and lp (p∈(0,1)) models, see Wang (2013). We also devise two DC Algorithms on the spirit of a paper where exact DC representation of the cardinality constraint is investigated and which also used the largest-q norm of lσq and presented numerical results that show the efficiency of our DC Algorithm in comparison with other methods using other penalty terms in the context of quadratic programing, see Jun-ya et al. (2017).

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Proximal methods for reweighted l-regularization of sparse signal recovery

Peng

2020

Applied Mathematics and Computation

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Properties and Iterative Methods for theQ-Lasso

Cited by 5 publications

References 13 publications

l 1-l 2 regularization of split feasibility problems

l 1-l 2 regularization of split feasibility problems

Difference of two norms-regularizations forQ-Lasso

Proximal methods for reweighted l-regularization of sparse signal recovery

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