l0伪随机数的一个合适共轭
A Suitable Conjugacy for the l0 Pseudonorm
摘要：
R d上的所谓l0计数向量的非零分量的数量。众所周知，l0的双共轭不是凸的，因为它的Fenchel双共轭是零。在本文中，我们引入了一个合适的共轭，诱导一种新的耦合，Caprac，具有沿原始射线是常数的性质，像l0的π。Caprac耦合属于我们介绍的单侧线性耦合类。我们表明，他们诱导共轭共享良好的性能与经典的Fenchel共轭。对于由耦合Caprac诱导的Caprac共轭，我们证明了l0的共轭性等于它的双共轭性：因此，l0的共轭性在广义凸意义下是Caprac-凸的.作为一个推论，我们表明，l0重合，在球面上，凸lsc函数。我们还提供了两个家庭的对偶规范，2-k-对称规范规范和k-支持规范的共轭表达式。
英文摘要：
The so-called l0 pseudonorm on R d counts the number of nonzero components of a vector. It is well-known that the l0 pseudonorm is not convex, as its Fenchel biconjugate is zero. In this paper, we introduce a suitable conjugacy, induced by a novel coupling, Caprac, having the property of being constant along primal rays, like the l0 pseudonorm. The Caprac coupling belongs to the class of one-sided linear couplings, that we introduce. We show that they induce conjugacies that share nice properties with the classic Fenchel conjugacy. For the Caprac conjugacy, induced by the coupling Caprac, we prove that the l0 pseudonorm is equal to its biconjugate: hence, the l0 pseudonorm is Caprac-convex in the sense of generalized convexity. As a corollary, we show that the l0 pseudonorm coincides, on the sphere, with a convex lsc function. We also provide expressions for conjugates in terms of two families of dual norms, the 2-k-symmetric gauge norms and the k-support norms.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
