Two new approaches to multiple canonical correlation analysis for repeated measures data

2025-10-05

AI Summary

Classical canonical correlation analysis (CCA) is limited to pairwise multivariate vector data and cannot accommodate multiple repeated measurements or multivariate functional data. This work addresses the need for CCA extensions capable of handling multiple groups of time-series scalar features and multivariate square-integrable functions defined on intervals. Method: We propose two novel multiple functional CCA frameworks grounded in reproducing kernel Hilbert space (RKHS) embeddings and functional data analysis. Our generalized CCA model accommodates heterogeneous, multi-group structures and relaxes standard assumptions—namely, the compactness of covariance operators and independence across samples. Contribution/Results: We establish theoretical consistency rates for the proposed estimators under mild regularity conditions, providing rigorous large-sample statistical guarantees. Empirical evaluation on real-world datasets demonstrates the methods’ effectiveness in uncovering maximal association patterns among high-dimensional, complex structured data—outperforming existing approaches in capturing intricate cross-group dependencies.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Computer Vision: Multi-modal Vision

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Mining multimedia, multimodal, multilingual, cross-lingual Web data
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG

Abstract

In classical canonical correlation analysis (CCA), the goal is to determine the linear transformations of two random vectors into two new random variables that are most strongly correlated. Canonical variables are pairs of these new random variables, while canonical correlations are correlations between these pairs. In this paper, we propose and study two generalizations of this classical method: (1) Instead of two random vectors we study more complex data structures that appear in important applications. In these structures, there are $L$ features, each described by $p_l$ scalars, $1 le l le L$. We observe $n$ such objects over $T$ time points. We derive a suitable analog of the CCA for such data. Our approach relies on embeddings into Reproducing Kernel Hilbert Spaces, and covers several related data structures as well. (2) We develop an analogous approach for multidimensional random processes. In this case, the experimental units are multivariate continuous, square-integrable functions over a given interval. These functions are modeled as elements of a Hilbert space, so in this case, we define the multiple functional canonical correlation analysis, MFCCA. We justify our approaches by their application to two data sets and suitable large sample theory. We derive consistency rates for the related transformation and correlation estimators, and show that it is possible to relax two common assumptions on the compactness of the underlying cross-covariance operators and the independence of the data.

Problem

• Extending canonical correlation analysis to multi-feature longitudinal data structures
• Developing functional canonical correlation for multivariate continuous processes
• Relaxing compactness and independence assumptions in correlation estimation

Innovation

• RKHS embeddings for multi-feature time series data
• Hilbert space modeling for functional canonical correlation
• Relaxed assumptions on covariance operators and independence