可约总最小二乘核心问题的酉不变分解
Unitary-Invariant Decomposition of Reducible Total Least Squares Core Problems
AI总结：
本文提出一种完全构造性框架，通过复域上的协方差算子谱结构，将总最小二乘核心问题精确分解为酉唯一不可约子问题，并证明其唯一性。
摘要：
总最小二乘问题（TLS）的分析可以简化为相关核心问题的分析，该核心问题通常具有更低的维度和更好的可解性。然而，即使核心问题也可能保持可约性，允许进一步分解为具有更简单结构和更好分析性质的不可约子问题。但是，无论是在实域还是复域上，目前还没有系统且不变的方法来识别所有这样的子问题。本文开发了一个完整且构造性的框架，用于将TLS核心问题精确分解为酉唯一不可约子问题。通过在复域上工作并利用与C-子集子问题相关的协方差算子的谱结构，所提出的策略产生了所有复不可分子空间，这些空间将导致不可约子问题。因此，我们证明了不可约子问题在酉变换和置换下是唯一确定的，从而部分解决了Yu, Jing在SIAM J. Matrix Anal. Appl., 46 (2025)中留下的一个开放问题。
英文摘要：
The analysis of a total least squares problem can be reduced to that of an associated core problem, which typically has lower dimension and improved solubility properties. Nevertheless, even a core problem may remain reducible, admitting further decomposition into irreducible component sub- problems with simpler structure and better analytical properties. We seek the finest decomposition under unitary transformations over C; for real data this decomposition can be strictly finer than the finest one under real orthogonal transformations. We split the data matrix into row blocks associated with the singular subspaces of the system matrix and consider the graph whose ver- tices are the blocks and whose edges correspond to nonzero cross-products. After preliminary splittings that make every nonzero cross-product a scalar multiple of a unitary matrix, we show that the components can be obtained directly from these blocks if and only if the covariance matrices along all cycles of the graph are scalar matrices. Otherwise, there is a chordless cycle whose covariance matrix is not a scalar matrix, and splitting the blocks on this cy- cle along the eigenspaces of the covariance matrix removes the obstruction without changing the decomposition. Based on these results, we propose a recursive strategy that finds all irreducible components over the complex field. The strategy is exact and theoretical rather than an efficient numerical algo- rithm; a small floating-point computation illustrates it. The results depend only on the core problem and therefore apply to general linear approximation problems AX B.
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