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arXiv 2007.15263math.NAcs.NA
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$α\ell_{1}-β\ell_{2}$稀疏正则化的投影梯度法

A projected gradient method for $α\ell_{1}-β\ell_{2}$ sparsity regularization

Liang Ding, Weimin Han

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30 pages; 8 figures
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摘要：

非凸$α\|\cdot\|_{\ell_1}-β\| \cdot\|_{\ell_2}$ $(α\geβ\geq0)$正则化在稀疏恢复领域引起了人们的关注。获得这种正则化的最小值的一种方法是ST-（$α\ell_1-β\ell_2$）算法，其类似于经典的迭代软阈值算法（ISTA）。众所周知，ISTA收敛速度非常慢，而ISTA的更快替代方法是投影梯度（PG）方法。然而，传统的PG方法仅限于经典的$\ell_1$稀疏正则化。在本文中，我们提出了两个加速替代ST-（$α\ell_1-β\ell_2$）算法通过扩展PG方法的非凸$α\ell_1-β\ell_2$稀疏正则化。此外，我们讨论了一种策略，以确定半径$R$的$\ell_1$ -球约束的Morozov的差异原则。数值结果表明，所提出的方法的效率。

英文摘要：

The non-convex $α\|\cdot\|_{\ell_1}-β\| \cdot\|_{\ell_2}$ $(α\geβ\geq0)$ regularization has attracted attention in the field of sparse recovery. One way to obtain a minimizer of this regularization is the ST-($α\ell_1-β\ell_2$) algorithm which is similar to the classical iterative soft thresholding algorithm (ISTA). It is known that ISTA converges quite slowly, and a faster alternative to ISTA is the projected gradient (PG) method. However, the conventional PG method is limited to the classical $\ell_1$ sparsity regularization. In this paper, we present two accelerated alternatives to the ST-($α\ell_1-β\ell_2$) algorithm by extending the PG method to the non-convex $α\ell_1-β\ell_2$ sparsity regularization. Moreover, we discuss a strategy to determine the radius $R$ of the $\ell_1$-ball constraint by Morozov's discrepancy principle. Numerical results are reported to illustrate the efficiency of the proposed approach.

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