Spectral operators of matrices
Spectral operators of matrices
复制标题
矩阵的谱算子
DOI：
10.1007/s10107-017-1162-3
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发表时间：
2018
影响因子：
2.7
通讯作者：
Toh Kim Chuan
中科院分区：
文献类型：
--
作者：
Ding Chao;Sun Defeng;Sun Jie;Toh Kim Chuan
近年来，矩阵优化问题(MOPS)已被公认为是建模许多涉及最优化领域内外的结构化低阶矩阵的重要应用的有力工具。这一趋势在一定程度上可以归功于压缩传感等新兴领域令人振奋的发展。L算子通过对矩阵的每个奇异值应用一个单变量函数来产生一个矩阵值函数，长期以来在求解矩阵优化问题中一直扮演着重要的角色。然而，为L算子发展起来的经典理论在最近的这些应用中已经变得不够充分。本文的主要目的是从设计求解多目标问题的有效数值方法的角度提供必要的理论基础。我们通过引入和深入研究一类新的矩阵值函数来实现这一目标，这类函数被称为矩阵的谱算子。系统地研究了谱算子的几个基本性质，包括适定性、连续性、方向可微性和Fréchet-可微性。
The class of matrix optimization problems (MOPs) has been recognized in recent years to be a powerful tool to model many important applications involving structured low rank matrices within and beyond the optimization community. This trend can be credited to some extent to the exciting developments in emerging fields such as compressed sensing. The Löwner operator, which generates a matrix valued function via applying a single-variable function to each of the singular values of a matrix, has played an important role for a long time in solving matrix optimization problems. However, the classical theory developed for the Löwner operator has become inadequate in these recent applications. The main objective of this paper is to provide necessary theoretical foundations from the perspectives of designing efficient numerical methods for solving MOPs. We achieve this goal by introducing and conducting a thorough study on a new class of matrix valued functions, coined as spectral operators of matrices. Several fundamental properties of spectral operators, including the well-definedness, continuity, directional differentiability and Fréchet-differentiability are systematically studied.