重新审视功能多组典型相关分析：从有限维样本到无限维总体
Functional Multiple-Set Canonical Correlation Analysis Revisited: From Finite-Dimensional Samples to Infinite-Dimensional Populations
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中文总结 AI 辅助
本文针对无限维希尔伯特空间的多变量功能数据，提出P-FMCCA总体形式，推导其良定条件与特征值刻画，建立典型分解，还探讨了功能同质性分析的有限维与无限维差异。
中文摘要 AI 辅助
我们针对无限维希尔伯特空间中的多变量功能数据，提出了功能多组典型相关分析的总体水平形式（P-FMCCA）。由于功能数据的协方差算子通常是紧的，因此在多组典型相关分析（MCCA）的形式扩展中出现的逆协方差算子一般是无界的，且未定义在整个希尔伯特空间上。因此，我们提供了确保所提出的总体形式良定的充分条件，并证明所得的约束最大化问题可由相关相关算子的希尔伯特-施密特扩展的特征值问题刻画。我们进一步建立了由P-FMCCA分量诱导的典型分解，并引入了相关的截断典型表示。此外，我们在总体水平上构建了功能同质性分析，并证明同质性分析与MCCA之间的有限维等价关系通常无法推广到无限维情形。最后，我们证明，对于有限秩截断典型表示，功能同质性分析可通过P-FMCCA分量获得显式刻画，从而为经典有限维对应关系提供了总体水平的对应物。
英文摘要
We develop a population-level formulation of functional multiple-set canonical correlation analysis (P-FMCCA) for multivariate functional data in an infinite-dimensional Hilbert space. Since covariance operators for functional data are typically compact, the inverse covariance operators that appear in the formal extension of multiple-set canonical correlation analysis (MCCA) are generally unbounded and are not defined on the whole Hilbert space. We therefore provide sufficient conditions under which the proposed population formulation is well-defined and show that the resulting constrained maximization problem is characterized by an eigenvalue problem for a Hilbert-Schmidt extension of the relevant correlation operator. We further establish a canonical decomposition induced by the P-FMCCA components and introduce the associated truncated canonical representation. In addition, we formulate functional homogeneity analysis at the population level and show that the finite-dimensional equivalence between homogeneity analysis and MCCA does not generally carry over to the infinite-dimensional setting. Finally, we prove that, for the finite-rank truncated canonical representation, functional homogeneity analysis admits an explicit characterization in terms of the P-FMCCA components, thereby providing a population-level counterpart of the classical finite-dimensional correspondence.
