A two-term penalty function for inverse problems with sparsity constrains. Rodriguez, P. In *2017 25th European Signal Processing Conference (EUSIPCO)*, pages 2126-2130, Aug, 2017. Paper doi abstract bibtex Inverse problems with sparsity constrains, such Basis Pursuit denoising (BPDN) and Convolutional BPDN (CBPDN), usually use the ℓ1-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 ℓ1-norm. In this paper we propose a two-term penalty function consisting of a synthesis between the ℓ1-norm and the penalty function associated with the Non-Negative Garrote (NNG) thresholding rule. Although the proposed two-term penalty function is non-convex, the total cost function for the BPDN / CBPDN problems is still convex. The performance of the proposed two-term 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.

@InProceedings{8081585,
author = {P. Rodriguez},
booktitle = {2017 25th European Signal Processing Conference (EUSIPCO)},
title = {A two-term penalty function for inverse problems with sparsity constrains},
year = {2017},
pages = {2126-2130},
abstract = {Inverse problems with sparsity constrains, such Basis Pursuit denoising (BPDN) and Convolutional BPDN (CBPDN), usually use the ℓ1-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 ℓ1-norm. In this paper we propose a two-term penalty function consisting of a synthesis between the ℓ1-norm and the penalty function associated with the Non-Negative Garrote (NNG) thresholding rule. Although the proposed two-term penalty function is non-convex, the total cost function for the BPDN / CBPDN problems is still convex. The performance of the proposed two-term 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.},
keywords = {approximation theory;convex programming;image coding;image denoising;inverse problems;optimisation;signal denoising;signal reconstruction;two-term penalty function;inverse problems;sparsity constrains;nonstandard penalty functions;BPDN-CBPDN problems;convolutional sparse coding problems;convolutional basis pursuit denoising;Signal processing algorithms;Convergence;Noise reduction;Convolutional codes;Convolution;Dictionaries;Europe},
doi = {10.23919/EUSIPCO.2017.8081585},
issn = {2076-1465},
month = {Aug},
url = {https://www.eurasip.org/proceedings/eusipco/eusipco2017/papers/1570346716.pdf},
}

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