论文标题
沿原始射线连接和L0伪型持续不断
Constant Along Primal Rays Conjugacies and the l0 Pseudonorm
论文作者
论文摘要
RD上所谓的L0伪型计数矢量的非零组分的数量。它用于稀疏优化，无论是标准还是在约束中，可获得很少的非零条目的解决方案。对于此类问题，Fenchel偶联性无法提供相关的分析：的确，L0伪型的水平集合集合的特征功能的Fenchel共轭是减无穷大，而L0 pseudonorm的Fenchel Biconjugate则是零。在本文中，我们展示了一类适合L0伪型的共轭。为此，我们假设在RD上给出了（源）规范。通过此规范，我们一方面定义了一个所谓的坐标-K规范的序列，另一方面是RD和RD之间的耦合，称为CAPRA（沿原始射线持续）。然后，我们为CAPRA缀合物和双缀合物以及CAPRA亚差异的公式提供了L0 pseudonorm的功能（尤其是L0 pseudonorm本身和其水平集集的特征函数）的函数，在Coorceard-k Norms中。作为应用程序，我们为L0伪型提供了一个新的较低界限，作为两个规范之间的一部分，分母是任何规范。
The so-called l0 pseudonorm on Rd counts the number of nonzero components of a vector. It is used in sparse optimization, either as criterion or in the constraints, to obtain solutions with few nonzero entries. For such problems, the Fenchel conjugacy fails to provide relevant analysis: indeed, the Fenchel conjugate of the characteristic function of the level sets of the l0 pseudonorm is minus infinity, and the Fenchel biconjugate of the l0 pseudonorm is zero. In this paper, we display a class of conjugacies that are suitable for the l0 pseudonorm. For this purpose, we suppose given a (source) norm on Rd. With this norm, we define, on the one hand, a sequence of so-called coordinate-k norms and, on the other hand, a coupling between Rd and Rd , called Capra (constant along primal rays). Then, we provide formulas for the Capra-conjugate and biconjugate, and for the Capra subdifferentials, of functions of the l0 pseudonorm (hence, in particular, of the l0 pseudonorm itself and of the characteristic functions of its level sets), in terms of the coordinate-k norms. As an application, we provide a new family of lower bounds for the l0 pseudonorm, as a fraction between two norms, the denominator being any norm.
