Joint modeling of high-dimensional longitudinal data and survival using supervised low-rank tensor decomposition.

High-dimensional longitudinal data are increasingly available in biomedical research, especially from omics platforms, but pose substantial challenges for joint modeling with survival outcomes. These challenges include modeling complex temporal dynamics, accommodating cross-feature dependencies, and maintaining computational feasibility. We propose a novel joint modeling framework that addresses these issues using supervised low-rank functional tensor decomposition to capture latent structure in multivariate longitudinal data and proportional hazards modeling for time-to-event outcomes. The longitudinal process is represented as a multivariate functional tensor, with a low-rank approximation that incorporates supervision from baseline covariates. Estimation is performed using a likelihood-based Monte Carlo Expectation-Maximization algorithm, enabling coherent inference and individualized prediction. Our method produces dynamic predictions of both longitudinal feature trajectories and survival probabilities. Simulation studies demonstrate substantial improvements in estimation accuracy and predictive performance over a standard two-stage approach, particularly under high censoring and limited sample sizes. In application to the Alzheimer's Disease Neuroimaging Initiative lipidomics data, the proposed model explains over 99% of variation with four components, and identifies significant subject-level latent predictors of dementia onset. This framework provides a scalable and interpretable strategy for integrating high-dimensional longitudinal biomarkers into joint models for disease progression and risk stratification.