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Capped $\ell_p$ approximations for the composite $\ell_0$ regularization problem

  • * Corresponding author: Na Zhang

    * Corresponding author: Na Zhang
The research is supported by the Natural Science Foundation of China under grants 11501584, 11626103 and 11701189, by the Natural Science Foundation of Guangdong Province under grants 2014A030310332 and 2014A030310414, and by the Fundamental Research Funds for the Central Universities of China.
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  • The composite $\ell_0$ function serves as a sparse regularizer in many applications. The algorithmic difficulty caused by the composite $\ell_0$ regularization (the $\ell_0$ norm composed with a linear mapping) is usually bypassed through approximating the $\ell_0$ norm. We consider in this paper capped $\ell_p$ approximations with $p>0$ for the composite $\ell_0$ regularization problem. The capped $\ell_p$ function converges to the $\ell_0$ norm pointwisely as the approximation parameter tends to infinity. We first establish the existence of optimal solutions to the composite $\ell_0$ regularization problem and its capped $\ell_p$ approximation problem under conditions that the data fitting function is asymptotically level stable and bounded below. Asymptotically level stable functions cover a rich class of data fitting functions encountered in practice. We then prove that the capped $\ell_p$ problem asymptotically approximates the composite $\ell_0$ problem if the data fitting function is a level bounded function composed with a linear mapping. We further show that if the data fitting function is the indicator function on an asymptotically linear set or the $\ell_0$ norm composed with an affine mapping, then the composite $\ell_0$ problem and its capped $\ell_p$ approximation problem share the same optimal solution set provided that the approximation parameter is large enough.

    Mathematics Subject Classification: Primary: 90C26, 49K40; Secondary: 49M30.


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  • Figure 1.  Capped $\ell_p$ function $\varphi_\gamma$ with $\gamma = 1$ for $p = 0.5, 1, 2$

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