Proof of Convergence and Performance Analysis for Sparse Recovery via Zero-Point Attracting Projection

Wang, Xiaohan; Gu, Yuantao; Chen, Laming

doi:10.1109/tsp.2012.2195660

Cited by 21 publications

(14 citation statements)

References 42 publications

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“…Hence, using (10) and 11the 1D ZALMS can be used as a 2D-ZALMS algorithm in (12) and the update equation becomes:…”

Section: Extending To the 2d Casementioning

confidence: 99%

Image denoising with two-dimensional zero attracting LMS algorithm

Eleyan¹,

Salman²

2019

Pamukkale J Eng Sci

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In this paper, we propose a new two-dimensional (2D) zero-attracting least-mean-square (ZALMS) adaptive filter by imposing a sparsity aware l1-norm penalty term into the cost function of the 2D-LMS algorithm. Comparisons with 2D-LMS and BM3D algorithms were conducted both on sparse and non-sparse images. The carried-out simulations show that the proposed algorithm has good capabilities in updating the filter coefficients along both horizontal and vertical directions, and its performance is similar with the 2D-LMS algorithm with lower computation time. But 2D-ZALMS performs better than BM3D algorithm. Bu yazıda, iki boyutlu en küçük kare algoritmasının (2D-LMS) maliyet fonksiyonuna seyrekliği farkeden l1-norm ceza terimi yükleyen yeni bir 2D sıfıra çeken en küçük ortalama kare (ZALMS) uyarlamalı filtreyi önermekteyiz. 2D-LMS ve BM3D algoritmaları ile karşılaştırmalar hem seyrek hem de seyrek olmayan görüntülerde yürütülmüştür. Simülasyon sonuçları, önerilen algoritmanın hem yatay hem de dikey doğrultuda filtre katsayılarının güncellenmesinde iyi yeteneklere sahip olduğunu göstermiştir ve performansı düşük hesaplama zamanına sahip 2D-LMS algoritması ile aynı/daha iyidir. Ancak 2D-ZALMS, BM3D algoritmasından daha iyi performans göstermektedir.

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“…Hence, using (10) and 11the 1D ZALMS can be used as a 2D-ZALMS algorithm in (12) and the update equation becomes:…”

Section: Extending To the 2d Casementioning

confidence: 99%

Image denoising with two-dimensional zero attracting LMS algorithm

Eleyan¹,

Salman²

2019

Pamukkale J Eng Sci

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show abstract

“…where U I was defined in (15). Clearly U c I is of measure zero and of the first category for each I , so is S c , a finite union of them.…”

Section: B Equivalence Regained: J \ R J Is Zero Measure and Meagrementioning

confidence: 99%

“…Although the non-convex nature of these cost functions makes it difficult to exactly solve the corresponding optimization problems, various practical algorithms can be adapted to these non-convex problems, including the iteratively re-weighted least squares minimization (IRLS) [12], [13], iterative thresholding algorithm (IT) [14], which are based on fixed point iteration; and the zero point attracting projection algorithm (ZAP) [3], [15], [16], which is based on Newton's method for solving nonlinear optimization. In general the non-convex algorithms have empirically outperformed BP in the various respects, because nonlinear cost functions can better promote sparsity than the 1 cost function.…”

Section: Introductionmentioning

confidence: 99%

Robustness of Sparse Recovery via <inline-formula> <tex-math notation="LaTeX">$F$ </tex-math></inline-formula>-Minimization: A Topological Viewpoint

Liu

Jin

2015

IEEE Trans. Inform. Theory

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A recent trend in compressed sensing is to consider nonconvex optimization techniques for sparse recovery. The important case of F-minimization has become of particular interest, for which the exact reconstruction condition (ERC) in the noiseless setting can be precisely characterized by the null space property (NSP). However, little work has been done concerning its robust reconstruction condition (RRC) in the noisy setting. We look at the null space of the measurement matrix as a point on the Grassmann manifold, and then study the relation between the ERC and RRC sets, denoted as J and r J , respectively. It is shown that r J is the interior of J , from which a previous result of the equivalence of ERC and RRC for p -minimization follows easily as a special case. Moreover, when F is nondecreasing, it is shown that J \ int( J ) is a set of measure zero and of the first category. As a consequence, the probabilities of ERC and RRC are the same if the measurement matrix A is randomly generated according to a continuous distri- bution. Quantitatively, if the null space N (A) lies in the d-interior of J , then RRC will be satisfied with the robustness constant C = 2 + 2d/dσ min (A ); and conversely, if RRC holds with C = 2 − 2d/dσ max (A ), then N (A) must lie in d-interior of J .We also present several rules for comparing the performances of different cost functions. Finally, these results are capitalized to derive achievable tradeoffs between the measurement rate and robustness with the aid of Gordon's escape through the mesh theorem or a connection between NSP and the restricted eigenvalue condition. He is the author/coauthor of two textbooks on Signals and Systems, and more than 50 papers on signal processing, multimedia communications, and wireless networks. His research interests include adaptive filtering, sparse signal recovery, multimedia signal processing, and related topics in wireless communications and information networks. Currently, he serves as Associate Editor for IEEE TRANSACTIONS ON SIGNAL PROCESSING and Handling Editor for EURASIP Digital Signal Processing.

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“…In many engineering and mathematics problems, sparsity is a popular topic [4]. In system identification and communication applications, the system may be in sparse nature [5] such as acoustic echo cancellation and network cancellation applications.…”

Section: Introductionmentioning

confidence: 99%

Image denoising with two-dimensional zero attracting LMS algorithm

Eleyan¹,

Salman²

2019

Pamukkale J Eng Sci

View full text Add to dashboard Cite

In this paper, we propose a new two-dimensional (2D) zero-attracting least-mean-square (ZALMS) adaptive filter by imposing a sparsity aware l1-norm penalty term into the cost function of the 2D-LMS algorithm. Comparisons with 2D-LMS and BM3D algorithms were conducted both on sparse and non-sparse images. The carried-out simulations show that the proposed algorithm has good capabilities in updating the filter coefficients along both horizontal and vertical directions, and its performance is similar with the 2D-LMS algorithm with lower computation time. But 2D-ZALMS performs better than BM3D algorithm. Bu yazıda, iki boyutlu en küçük kare algoritmasının (2D-LMS) maliyet fonksiyonuna seyrekliği farkeden l1-norm ceza terimi yükleyen yeni bir 2D sıfıra çeken en küçük ortalama kare (ZALMS) uyarlamalı filtreyi önermekteyiz. 2D-LMS ve BM3D algoritmaları ile karşılaştırmalar hemseyrek hem de seyrek olmayan görüntülerde yürütülmüştür. Simülasyon sonuçları, önerilen algoritmanın hem yatay hem de dikey doğrultuda filtre katsayılarının güncellenmesinde iyi yeteneklere sahip olduğunu göstermiştir ve performansı düşük hesaplama zamanına sahip 2D-LMS algoritması ile aynı/daha iyidir. Ancak 2D-ZALMS, BM3D algoritmasından daha iyi performans göstermektedir.

show abstract

Proof of Convergence and Performance Analysis for Sparse Recovery via Zero-Point Attracting Projection

Cited by 21 publications

References 42 publications

Image denoising with two-dimensional zero attracting LMS algorithm

Image denoising with two-dimensional zero attracting LMS algorithm

Robustness of Sparse Recovery via <inline-formula> <tex-math notation="LaTeX">$F$ </tex-math></inline-formula>-Minimization: A Topological Viewpoint

Image denoising with two-dimensional zero attracting LMS algorithm

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