Functional Multiple-Set Canonical Correlation Analysis Revisited: From Finite-Dimensional Samples to Infinite-Dimensional Populations

Michio Yamamoto; Yoshikazu Terada

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arXiv · 2608.22170

Functional Multiple-Set Canonical Correlation Analysis Revisited: From Finite-Dimensional Samples to Infinite-Dimensional Populations

Michio Yamamoto·Yoshikazu Terada

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Abstract

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.

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Michio Yamamoto, Yoshikazu Terada. 2026-08-23. Functional Multiple-Set Canonical Correlation Analysis Revisited: From Finite-Dimensional Samples to Infinite-Dimensional Populations. https://arxiv.org/abs/2608.22170

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