Latent Functional PARAFAC for modeling multidimensional longitudinal data

In numerous settings, it is increasingly common to deal with longitudinal data organized as high-dimensional multi-dimensional arrays, also known as tensors. Within this framework, the time-continuous property of longitudinal data often implies a smooth functional structure on one of the tensor modes. To help researchers investigate such data, we introduce a new tensor decomposition approach based on the CANDECOMP/PARAFAC decomposition. Our approach allows for representing a high-dimensional functional tensor as a low-dimensional set of functions and feature matrices. Furthermore, to capture the underlying randomness of the statistical setting more efficiently, we introduce a probabilistic latent model in the decomposition. A covariance-based block-relaxation algorithm is derived to obtain estimates of model parameters. Thanks to the covariance formulation of the solving procedure and thanks to the probabilistic modeling, the method can be used in sparse and irregular sampling schemes, making it applicable in numerous settings. We apply our approach to help characterize multiple neurocognitive scores observed over time in the Alzheimer’s Disease Neuroimaging Initiative (ADNI) study. Finally, intensive simulations show a notable advantage of our method in reconstructing tensors.

💡 Research Summary

The paper addresses the growing need to analyze longitudinal data that are naturally represented as high‑dimensional tensors, where one mode (typically time) exhibits a smooth functional structure and observations are often collected irregularly and sparsely. Classical tensor decompositions such as CANDECOMP/PARAFAC (CP) provide low‑rank representations but ignore functional continuity and the stochastic nature of the sampling process. To fill this gap, the authors propose a novel framework called Latent Functional PARAFAC (LF‑PARAFAC), which integrates functional tensor decomposition with a probabilistic latent model for the sample mode.

The methodology begins by extending the CP model to functional tensors. For a tensor of order D, each rank‑1 component is multiplied by a scalar function ϕₖ(t) defined on a time interval I, yielding the functional PARAFAC (F‑PARAFAC) representation:
X(t) ≈ Σₖ ϕₖ(t)·aₖ^{(1)} ⊙ … ⊙ aₖ^{(D)}.
To capture randomness inherent in longitudinal sampling, a scalar latent variable uₖ is introduced for each component, leading to the final LF‑PARAFAC model:
X(t) ≈ Σₖ uₖ·ϕₖ(t)·aₖ^{(1)} ⊙ … ⊙ aₖ^{(D)}.
The latent variables are assumed to follow a simple distribution (e.g., Gaussian), enabling a Bayesian interpretation and individual‑specific inference.

Parameter estimation is performed via a covariance‑based block‑relaxation algorithm. Holding the functional factors Φ = {ϕₖ} and the factor matrices A^{(d)} fixed, the optimal operator Ψ* that maps a tensor to its latent sample‑mode vector is derived in closed form using cross‑covariance operators between the observed tensor and the current factors. This yields an explicit expression for the covariance matrix Λ* = E