Improved Iteratively Reweighted Least Squares for Unconstrained Smoothed Lq Minimization

Ming-Jun Lai Yangyang Xu Wotao Yin

Numerical Linear Algebra mathscidoc:1912.43140

SIAM Journal on Numerical Analysis, 51, (2), 927-957, 2013.3
In this paper, we first study \ell_q minimization and its associated iterative reweighted algorithm for recovering sparse vectors. Unlike most existing work, we focus on <i>unconstrained</i> \ell_q minimization, for which we show a few advantages on noisy measurements and/or approximately sparse vectors. Inspired by the results in [Daubechies et al., <i>Comm. Pure Appl. Math.</i>, 63 (2010), pp. 1--38] for <i>constrained</i> \ell_q minimization, we start with a preliminary yet novel analysis for <i>unconstrained</i> \ell_q minimization, which includes convergence, error bound, and local convergence behavior. Then, the algorithm and analysis are extended to the recovery of low-rank matrices. The algorithms for both vector and matrix recovery have been compared to some state-of-the-art algorithms and show superior performance on recovering sparse vectors and low-rank matrices.
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@inproceedings{ming-jun2013improved,
  title={Improved Iteratively Reweighted Least Squares for Unconstrained Smoothed Lq Minimization},
  author={Ming-Jun Lai, Yangyang Xu, and Wotao Yin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221112514240744700},
  booktitle={SIAM Journal on Numerical Analysis},
  volume={51},
  number={2},
  pages={927-957},
  year={2013},
}
Ming-Jun Lai, Yangyang Xu, and Wotao Yin. Improved Iteratively Reweighted Least Squares for Unconstrained Smoothed Lq Minimization. 2013. Vol. 51. In SIAM Journal on Numerical Analysis. pp.927-957. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221112514240744700.
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