Optimal Transport based Cross-Domain Integration for Heterogeneous Data

Detecting dynamic patterns shared across heterogeneous datasets is a critical yet challenging task in many scientific domains, particularly within the biomedical sciences. Systematic heterogeneity inherent in diverse data sources can significantly hinder the effectiveness of existing machine learning methods in uncovering shared underlying dynamics. Additionally, practical and technical constraints in real-world experimental designs often limit data collection to only a small number of subjects, even when rich, time-dependent measurements are available for each individual. These limited sample sizes further diminish the power to detect common dynamic patterns across subjects. In this paper, we propose a novel heterogeneous data integration framework based on optimal transport to extract shared patterns in the conditional mean dynamics of target responses. The key advantage of the proposed method is its ability to enhance discriminative power by reducing heterogeneity unrelated to the signal. This is achieved through the alignment of extracted domain-shared temporal information across multiple datasets from different domains. Our approach is effective regardless of the number of datasets and does not require auxiliary matching information for alignment. Specifically, the method aligns longitudinal data from heterogeneous datasets within a common latent space, capturing shared dynamic patterns while leveraging temporal dependencies within subjects. Theoretically, we establish generalization error bounds for the proposed data integration approach in supervised learning tasks, highlighting a novel trade-off between data alignment and pattern learning. Additionally, we derive convergence rates for the barycentric projection under Gromov-Wasserstein and fused Gromov-Wasserstein distances. Numerical studies on both simulated data and neuroscience applications demonstrate that the proposed data integration framework substantially improves prediction accuracy by effectively aggregating information across diverse data sources and subjects.