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Associating High-Dimensional Longitudinal Datasets through an Efficient Cross-Covariance Decomposition

Jianbin Tan, Pixu Shi

TL;DR

FACD addresses the challenge of identifying time-varying associations between paired high-dimensional longitudinal datasets by decomposing their cross-covariance operators with data-adaptive bases, reducing the problem to a tractable sparse matrix SVD. The method yields canonical loadings and scores that capture dynamic cross-view relationships while accommodating irregular sampling and enabling feature selection. The authors provide consistency guarantees in high-dimensional regimes, demonstrate superior recovery and variable selection in simulations, and validate the approach on a longitudinal multi-omic exercise study, uncovering biologically meaningful time-varying cross-omic associations. Collectively, FACD advances dynamic multi-omics integration and offers a scalable framework for probing coordinated biological processes over time.

Abstract

Understanding associations between paired high-dimensional longitudinal datasets is a fundamental yet challenging problem that arises across scientific domains, including longitudinal multi-omic studies. The difficulty stems from the complex, time-varying cross-covariance structure coupled with high dimensionality, which complicates both model formulation and statistical estimation. To address these challenges, we propose a new framework, termed Functional-Aggregated Cross-covariance Decomposition (FACD), tailored for canonical cross-covariance analysis between paired high-dimensional longitudinal datasets through a statistically efficient and theoretically grounded procedure. Unlike existing methods that are often limited to low-dimensional data or rely on explicit parametric modeling of temporal dynamics, FACD adaptively learns temporal structure by aggregating signals across features and naturally accommodates variable selection to identify the most relevant features associated across datasets. We establish statistical guarantees for FACD and demonstrate its advantages over existing approaches through extensive simulation studies. Finally, we apply FACD to a longitudinal multi-omic human study, revealing blood molecules with time-varying associations across omic layers during acute exercise.

Associating High-Dimensional Longitudinal Datasets through an Efficient Cross-Covariance Decomposition

TL;DR

FACD addresses the challenge of identifying time-varying associations between paired high-dimensional longitudinal datasets by decomposing their cross-covariance operators with data-adaptive bases, reducing the problem to a tractable sparse matrix SVD. The method yields canonical loadings and scores that capture dynamic cross-view relationships while accommodating irregular sampling and enabling feature selection. The authors provide consistency guarantees in high-dimensional regimes, demonstrate superior recovery and variable selection in simulations, and validate the approach on a longitudinal multi-omic exercise study, uncovering biologically meaningful time-varying cross-omic associations. Collectively, FACD advances dynamic multi-omics integration and offers a scalable framework for probing coordinated biological processes over time.

Abstract

Understanding associations between paired high-dimensional longitudinal datasets is a fundamental yet challenging problem that arises across scientific domains, including longitudinal multi-omic studies. The difficulty stems from the complex, time-varying cross-covariance structure coupled with high dimensionality, which complicates both model formulation and statistical estimation. To address these challenges, we propose a new framework, termed Functional-Aggregated Cross-covariance Decomposition (FACD), tailored for canonical cross-covariance analysis between paired high-dimensional longitudinal datasets through a statistically efficient and theoretically grounded procedure. Unlike existing methods that are often limited to low-dimensional data or rely on explicit parametric modeling of temporal dynamics, FACD adaptively learns temporal structure by aggregating signals across features and naturally accommodates variable selection to identify the most relevant features associated across datasets. We establish statistical guarantees for FACD and demonstrate its advantages over existing approaches through extensive simulation studies. Finally, we apply FACD to a longitudinal multi-omic human study, revealing blood molecules with time-varying associations across omic layers during acute exercise.
Paper Structure (14 sections, 5 theorems, 28 equations, 6 figures, 1 algorithm)

This paper contains 14 sections, 5 theorems, 28 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Functional-Aggregated Cross-covariance Decomposition
  3. Canonical Cross-Covariance between Functional Data
  4. Functional-Aggregated Decomposition of Cross-Covariances
  5. Canonical Cross-Covariance Analysis on Observed Data
  6. Basis Function Estimation
  7. Estimation of Canonical Functional Loadings and Scores
  8. Statistical Consistency
  9. Simulation
  10. Data Generation
  11. Competing Methods and Evaluation Measures
  12. Simulation Results
  13. Data Analysis
  14. Discussion

Key Result

Theorem 1

Assume that $\bm{R}_{XY}(t_1, t_2)$ is continuous with respect to $t_1$ and $t_2$. Then, $\mathcal{R}_{XY}$ is a compact operator with the following decomposition: where $\lambda_1\geq \lambda_2\geq \cdots\geq 0$ are the singular values, and $\widetilde{\bm{A}}^X_{r}$ and $\widetilde{\bm{A}}^Y_{r}$ are singular functions, respectively. In addition, where the maximum is attained at $\bm{A}_X=\pm\

Figures (6)

  • Figure 1: Diagram illustrating longitudinal associations between two high-dimensional longitudinal datasets. Each dataset is represented by an array indexed by feature, time and subject. Yellow arrows indicate functional loadings that aggregate features in a time-dependent manner, red circles denote aggregated features whose cross-covariance between datasets is maximized, and missing observations at certain time points are shown in gray.
  • Figure 2: Boxplot of estimation errors for functional loadings from the first (Panel A) and second (Panel B) datasets.
  • Figure 3: The false positive rates and false negative rates of variable selection.
  • Figure 4: (A) The extracted first canonical scores of the participants between any two omics types of high-dimensional longitudinal data, with correlations calculated using the Pearson correlation coefficient. (B) Boxplot of Pearson correlations between the scores of proteome and metabolome for each testing fold.
  • Figure 5: The estimated functional loadings between any two omics of high-dimensional longitudinal data. The signs of the functional loadings are determined such that their inner products with the constant function are positive. The signs of lipid features depend on the signs used in the principal component (PC) analysis, where the PC loadings are arranged to have a positive inner product with the constant vector. We only present the top 10 important features ranked by the $L^2$ norm of their functional loadings.
  • ...and 1 more figures

Theorems & Definitions (6)

  • Theorem 1
  • Remark 1: Challenges of Canonical Cross-Covariance Analysis in High Dimensions
  • Theorem 2: Basis Expansion of Cross-Covariance
  • Theorem 3: Extraction of Functional Bases
  • Theorem 4: Perturbation Bound
  • Theorem 5