Computing Sparse Representation in a Highly Coherent Dictionary Based on Difference of L_1 and L_1L2

Yifei Lou Penghang Yin Qi He Jack Xin

Numerical Analysis and Scientific Computing mathscidoc:1912.43845

Journal of Scientific Computing, 64, (1), 178-196, 2015.7
We study analytical and numerical properties of the L 1 - L 2 minimization problem for sparse representation of a signal over a highly coherent dictionary. Though the L 1 - L 2 metric is non-convex, it is Lipschitz continuous. The difference of convex algorithm (DCA) is readily applicable for computing the sparse representation coefficients. The L 1 - L 2 minimization appears as an initialization step of DCA. We further integrate DCA with a non-standard simulated annealing methodology to approximate globally sparse solutions. Non-Gaussian random perturbations are more effective than standard Gaussian perturbations for improving sparsity of solutions. In numerical experiments, we conduct an extensive comparison among sparse penalties such as L 1 - L 2 for L 1 - L 2 based on data from three specific applications (over-sampled discreet cosine basis, differential absorption optical spectroscopy, and image denoising) where
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@inproceedings{yifei2015computing,
  title={Computing Sparse Representation in a Highly Coherent Dictionary Based on Difference of L_1 and L_1L2},
  author={Yifei Lou, Penghang Yin, Qi He, and Jack Xin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224210145124438409},
  booktitle={Journal of Scientific Computing},
  volume={64},
  number={1},
  pages={178-196},
  year={2015},
}
Yifei Lou, Penghang Yin, Qi He, and Jack Xin. Computing Sparse Representation in a Highly Coherent Dictionary Based on Difference of L_1 and L_1L2. 2015. Vol. 64. In Journal of Scientific Computing. pp.178-196. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224210145124438409.
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