论文标题
Quaternion矩阵优化和基础演算
Quaternion Matrix Optimization and The Underlying Calculus
论文作者
论文摘要
涉及四个矩阵的优化模型被广泛用于颜色图像过程和其他工程领域。这些模型优化了四个矩阵变量的实际功能。特别是，$ \ ell_0 $ norms和Quaternion矩阵的等级函数是离散的。然而，具有这种实际功能的衍生物，细分和广义的细分差分来处理此类模型，需要进行微积分。在本文中，我们介绍了一阶和二阶导数，并为这种实际功能建立了他们的计算规则。我们的方法与最近在文献中引入的四元矩阵变量规范的亚级别概念一致。我们开发了Quaternion矩阵正常功能的广义细分概念的概念，并使用它们来分析稀疏的低级颜色图像DeNoisis模型的最佳条件。我们引入了两个四元矩阵向量的R-产品，作为我们的微积分的关键工具。我们表明，低级别四元素矩阵的真实表示集是封闭和半代数的。我们还建立了二阶和二阶最优条件，以解决季度矩阵变量中实际功能的约束优化问题。
Optimization models involving quaternion matrices are widely used in color image process and other engineering areas. These models optimize real functions of quaternion matrix variables. In particular, $\ell_0$-norms and rank functions of quaternion matrices are discrete. Yet calculus with derivatives, subdifferentials and generalized subdifferentials of such real functions is needed to handle such models. In this paper, we introduce first and second order derivatives and establish their calculation rules for such real functions. Our approach is consistent with the subgradient concept for norms of quaternion matrix variables, recently introduced in the literature. We develop the concepts of generalized subdifferentials of proper functions of quaternion matrices, and use them to analyze the optimality conditions of a sparse low rank color image denoising model. We introduce R-product for two quaternion matrix vectors, as a key tool for our calculus. We show that the real representation set of low-rank quaternion matrices is closed and semi-algebraic. We also establish first order and second order optimality conditions for constrained optimization problems of real functions in quaternion matrix variables.
