The Proximity Operator of the Log-Sum Penalty
The Proximity Operator of the Log-Sum Penalty
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DOI：
10.1007/s10915-022-02021-4
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发表时间：
2022-12-01
影响因子：
2.5
通讯作者：
Tripp，Erin E.
中科院分区：
文献类型：
--
作者：
Prater-Bennette，Ashley;Shen，Lixin;Tripp，Erin E.
在压缩感知和低秩优化中，对数和惩罚常被用来代替伪范数。惩罚的接近算子，即硬阈值算子，在应用中起着至关重要的作用；同样，我们需要一种有效的方法来评估对数和惩罚的接近算子。由于该函数的非凸性，其接近算子通常通过迭代加权方法计算，该方法将对数和项替换为其一阶近似值。本文报道了对数和惩罚的接近算子实际上有一个显式表达式。利用它，我们证明了迭代加权解在某些区域与真邻近算子不一致。作为一个副产品，迭代重加权解被精确地表征为所选择的初始化。我们还给出了对数和惩罚与奇异值函数组合的接近算子的显式形式，如在低秩应用中所见。这些结果应该对开发涉及对数和惩罚的优化问题的高效和准确算法有用。我们提出了解决压缩感知问题和混合加性高斯白噪声和脉冲噪声去除的应用。
The log-sum penalty is often adopted as a replacement for thepseudo-norm in compressive sensing and low-rank optimization. The proximity operator of thepenalty, i.e., the hard-thresholding operator, plays an essential role in applications; similarly, we require an efficient method for evaluating the proximity operator of the log-sum penalty. Due to the nonconvexity of this function, its proximity operator is commonly computed through the iteratively reweightedmethod, which replaces the log-sum term with its first-order approximation. This paper reports that the proximity operator of the log-sum penalty actually has an explicit expression. With it, we show that the iteratively reweightedsolution disagrees with the true proximity operator in certain regions. As a by-product, the iteratively reweightedsolution is precisely characterized in terms of the chosen initialization. We also give the explicit form of the proximity operator for the composition of the log-sum penalty with the singular value function, as seen in low-rank applications. These results should be useful in the development of efficient and accurate algorithms for optimization problems involving the log-sum penalty. We present applications to solving compressive sensing problems and to mixed additive Gaussian white noise and impulse noise removal.