论文标题
Schatten矩阵规范的二元映射
Duality Mapping for Schatten Matrix Norms
论文作者
论文摘要
在本文中，我们充分表征了配备了Schatten规范的矩阵空间上的二元映射。我们的方法是基于对沙顿规范Hölder不平等饱和度的分析。我们在主要结果中证明，对于$ p \ in（1，\ infty）$，带有schatten-$ p $ norm的实价矩阵空间上的二重性映射是连续且单值函数，并为其计算提供了明确的表单。对于特殊情况$ p = 1 $，映射是设置值的；通过添加等级约束，我们表明它可以将其简化为可释放的单值函数，我们还可以为其提供封闭形式的表达式。
In this paper, we fully characterize the duality mapping over the space of matrices that are equipped with Schatten norms. Our approach is based on the analysis of the saturation of the Hölder inequality for Schatten norms. We prove in our main result that, for $p\in (1,\infty)$, the duality mapping over the space of real-valued matrices with Schatten-$p$ norm is a continuous and single-valued function and provide an explicit form for its computation. For the special case $p = 1$, the mapping is set-valued; by adding a rank constraint, we show that it can be reduced to a Borel-measurable single-valued function for which we also provide a closed-form expression.
