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Latent Functional PARAFAC for modeling multidimensional longitudinal data

This paper introduces Latent Functional PARAFAC, a probabilistic tensor decomposition method that models high-dimensional longitudinal data with smooth functional structures to enable robust analysis under sparse and irregular sampling schemes, as demonstrated through simulations and an application to Alzheimer's disease neurocognitive data.

Original authors: Lucas Sort, Laurent Le Brusquet, Arthur Tenenhaus

Published 2026-01-28

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CC BY 4.0

Original authors: Lucas Sort, Laurent Le Brusquet, Arthur Tenenhaus

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to understand a massive, messy library of books. But these aren't normal books; they are living, breathing stories that change every day, written by thousands of different people, and some pages are missing or torn out.

This is the problem the authors of this paper are trying to solve. They are dealing with longitudinal data—information collected over time (like a patient's health scores measured every few months for ten years). But instead of just one number per person, they have a whole "tensor" (a multi-dimensional data cube) containing many different scores, measured at different times, for many different people.

Here is the simple breakdown of their solution, using everyday analogies:

1. The Problem: The "Messy Library"

In the real world, data is rarely perfect.

It's Continuous: Time flows smoothly, like a river, but we only take snapshots (measurements) at random moments.
It's Irregular: Person A might be measured in January, March, and October. Person B might be measured in February and July.
It's Noisy: The measurements aren't perfect; there's always some "static" or error.
It's Huge: If you have 1,000 people, 10 years of data, and 6 different tests, the data is too big to look at directly. It's like trying to read a million books at once.

Existing tools (called "Tensor Decomposition") are good at summarizing data, but they treat time like a list of separate, disconnected points. They don't understand that time is a smooth flow. They also struggle when data is missing or measured at weird times.

2. The Solution: The "Smart Summarizer" (LF-PARAFAC)

The authors invented a new tool called Latent Functional PARAFAC. Think of this as a super-smart summarizer that understands two special rules:

Rule A: The Smooth Story (Functional): Instead of treating time as a list of dots, this tool sees time as a smooth curve. It assumes that if you know the score at day 1 and day 3, you can guess the score at day 2 because the story flows smoothly. This allows it to fill in the gaps even if the data is sparse (missing pages).
Rule B: The Hidden Randomness (Probabilistic): The tool acknowledges that every person is different. It separates the "general story" (the average trend) from the "individual quirks" (random noise). It treats the data as if it were drawn from a giant bag of possibilities, allowing it to handle the natural randomness of human behavior.

3. How It Works: The "Recipe"

Imagine you want to describe a complex dish (the data) using only a few basic ingredients (the model).

The Ingredients (The Decomposition): The tool breaks the massive data cube down into three simple parts:
- The "Time Flavor" (Functional Mode): A smooth curve showing how things generally change over time (e.g., "cognitive decline gets steeper after year 5").
- The "Test Flavor" (Feature Mode): A list showing which tests are related. For example, it might realize that "Memory Test A" and "Memory Test B" always move together, so they share an ingredient.
- The "Person Flavor" (Sample Mode): A score for each person showing how much they follow the general trend versus how much they are an outlier.
The Cooking Process (The Algorithm):
The tool uses a "block relaxation" method. Imagine trying to solve a jigsaw puzzle where you can't see the whole picture. You fix one piece, then the next, then the next, over and over, until the picture snaps into place.
- Because the tool uses covariance (a mathematical way of measuring how things move together) rather than just raw numbers, it can cook this recipe even if the kitchen is messy (missing data) or the ingredients are scattered (irregular timing).

4. The Real-World Test: Alzheimer's Study

The authors tested their new tool on a real dataset from the Alzheimer's Disease Neuroimaging Initiative (ADNI).

The Data: They looked at 888 patients (some healthy, some with Alzheimer's) over 10 years. They tracked 6 different cognitive scores (like memory tests and daily living skills).
The Mess: The data was a nightmare. Patients were tested at different times, with many gaps. A standard tool failed completely because the data was too messy and high-dimensional.
The Result: Their new tool worked perfectly. It found 4 main "stories" (patterns) hidden in the data:
1. The Sharp Decline: A pattern showing a rapid drop in memory and function, mostly seen in Alzheimer's patients in the first 5 years.
2. The Stability: A pattern showing people who stayed healthy and stable over the decade.
3. The Slow Fade: A pattern showing a very gradual, slow decline.
4. The Twin Tests: A pattern showing how two specific memory tests (MOCA and MMSE) are so similar they basically tell the same story.

Crucially, the tool could clearly separate the "Healthy" group from the "Alzheimer's" group based on these patterns, even with all the missing data points.

5. The Simulation: The "Fake Data" Check

To prove it wasn't just luck, they created 100 fake datasets with known answers. They intentionally deleted 20%, 50%, and even 80% of the data points to make it very hard.

The Result: Their method was excellent at reconstructing the original data, even when 80% of the information was missing. It proved that their "Smooth + Random" approach is much better at handling messy, real-world data than older methods.

Summary

In short, the authors built a mathematical microscope that can look at messy, incomplete, time-based data and find the smooth, hidden stories underneath. It's like taking a blurry, fragmented video of a moving object and using math to reconstruct the clear, smooth motion of the object, even if the camera only caught a few frames. They showed it works great for tracking how people's minds change over time, specifically in the context of Alzheimer's disease.

Technical Summary: Latent Functional PARAFAC for Modeling Multidimensional Longitudinal Data

1. Problem Statement

The paper addresses the challenge of analyzing high-dimensional longitudinal data organized as tensors, where one mode (typically time) exhibits a smooth, continuous functional structure, and another mode (the sample mode) represents random draws from an underlying distribution.

Existing methods face specific limitations in this context:

Standard CPD (CANDECOMP/PARAFAC): While effective for dimensionality reduction, standard CPD treats entries as discrete values. It fails to integrate the inherent continuous structure of functional data and cannot handle irregular or sparse sampling schemes common in longitudinal studies.
Functional Data Analysis (FDA): Methods like Functional Principal Component Analysis (FPCA) handle smoothness but are often limited to vector-valued data or require vectorizing tensors, which destroys tensor structure and becomes computationally infeasible for high-order tensors.
Existing Tensor-Functional Hybrids: Previous attempts to combine these fields (e.g., Guan [2023], Han et al. [2023]) are often restricted to specific tensor orders (e.g., order-3), lack probabilistic modeling to account for sampling randomness, or cannot handle irregular sampling designs.

The authors identify a gap for a general framework capable of decomposing functional tensors of any order, accommodating sparse and irregular sampling, and explicitly modeling the inherent randomness of the sample mode.

2. Methodology

2.1 The Latent Functional PARAFAC (LF-PARAFAC) Model

The authors propose a new decomposition model that extends the rank- $R$ CPD to functional tensors in a Hilbert space $\mathcal{H} = L^2(I)^{p_1 \times \dots \times p_D}$ .

The model represents a random functional tensor $\mathbf{X}$ as:
$\mathbf{X}(t) \approx \sum_{r=1}^R u_r \phi_r(t) (\mathbf{a}_{1r} \circ \dots \circ \mathbf{a}_{Dr})$
Where:

$\phi_r(t) \in L^2(I)$ are smooth functional components (basis functions) for the continuous mode.
$\mathbf{a}_{dr} \in \mathbb{R}^{p_d}$ are fixed factor matrices for the discrete modes.
$u_r$ are random variables representing the sample-mode scores, drawn from a distribution (typically Gaussian) to capture the stochastic nature of the data.
$\mathbf{U} = (u_1, \dots, u_R)$ is the vector of latent random scores.

This formulation, termed LF-PARAFAC, integrates the smoothness of functional data with the probabilistic structure of tensor decomposition.

2.2 Estimation via Covariance-Based Block Relaxation

To estimate the parameters ( $\Phi, \mathbf{A}_1, \dots, \mathbf{A}_D, \Lambda$ ), the authors derive a covariance-based block-relaxation algorithm (Algorithm 1).

Objective: The optimization minimizes the expected squared error between the observed tensor and the model. By integrating out the random variable $\mathbf{U}$ , the problem is reformulated to depend only on the covariance structure of the data.
Covariance Surfaces: The algorithm relies on estimating collapsed functional covariance matrices, $\Sigma^{[f]}(s,t)$ and $\Sigma^{[d]}(s,t)$ . These are derived from the cross-covariance surfaces of the functional entries.
Handling Irregularity: Following the approach of Yao et al. [2005], the method estimates these covariance surfaces using local linear smoothing on a regular grid. This allows the method to handle sparse and irregular observation times without requiring data imputation or alignment.
Iterative Procedure: The algorithm alternates between updating the covariance matrix of the latent scores ( $\Lambda$ ), the functional components ( $\Phi$ ), and the discrete factor matrices ( $\mathbf{A}_d$ ) until convergence.

2.3 Sample-Mode Inference (Bayesian Approach)

For a specific subject with sparse observations, directly estimating the latent scores $\mathbf{U}$ via least squares can be unstable. The authors propose a Bayesian estimator (Proposition 4) assuming a joint Gaussian distribution for the latent scores and residual errors. This yields an empirical Bayes estimator that borrows strength from the population covariance structure to predict individual trajectories, even with very few observations.

2.4 Identifiability

The paper establishes conditions for the identifiability of the LF-PARAFAC model (Proposition 3). Under scaling and permutation constraints, the model is identifiable if the functional space is infinite-dimensional and the $k$ -ranks of the factor matrices satisfy a specific inequality similar to standard CPD conditions.

3. Key Contributions

General Framework: Introduction of the LF-PARAFAC model, which extends CPD to functional tensors of any order while incorporating a probabilistic latent structure.
Robustness to Sampling: Development of a covariance-based estimation procedure that naturally handles sparse and irregular sampling schemes, a critical feature for longitudinal data where time points vary across subjects.
Probabilistic Integration: Explicit modeling of the sample mode as a random distribution, allowing for more efficient capture of underlying randomness and enabling Bayesian inference for individual subjects.
Algorithmic Solution: Derivation of a block-relaxation algorithm that utilizes cross-covariance surfaces, making the decomposition computationally feasible for high-dimensional, sparse data.

4. Results

4.1 Application to ADNI Data

The method was applied to the Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset, analyzing 6 neurocognitive scores (ADAS, MMSE, RAVLT, FAQ, MOCA, LDEL) over 10 years for 888 subjects (518 Cognitively Normal, 370 Alzheimer's Disease).

Data Characteristics: The data featured highly irregular sampling (1064 unique time points) and missing values, causing standard PARAFAC to fail.
Findings: A rank-4 decomposition successfully captured distinct trajectories:
- Component 1: Captured sharp cognitive deterioration over the first 5 years, clearly separating AD patients (high deterioration scores) from Cognitively Normal (CN) patients.
- Component 2: Described stable cognitive measures over the decade.
- Component 3: Represented a slow, gradual decline.
- Component 4: Captured shared variance between MOCA and MMSE scores.
Interpretability: The functional components provided smooth, interpretable trajectories, and the latent scores effectively stratified patients by diagnosis.

4.2 Simulation Study

Simulations were conducted on synthetic tensors of ranks 3, 5, and 10 with varying sparsity levels (0% to 80% missing data).

Performance: The simulations demonstrated a "notable advantage" of the LF-PARAFAC method in reconstructing tensors compared to existing approaches, particularly under high sparsity and irregular sampling conditions.
Robustness: The method maintained accuracy in recovering the underlying functional and discrete structures even when a large proportion of observations were removed.

5. Significance and Claims

The paper claims that the LF-PARAFAC approach provides a necessary extension to existing tensor and functional data analysis tools. Its primary significance lies in its ability to:

Unify smooth functional structures with probabilistic tensor decomposition.
Operate effectively in realistic longitudinal settings characterized by sparsity and irregularity, where traditional tensor methods fail.
Enhance interpretability by providing smooth functional components and probabilistic scores that can be linked to covariates (e.g., diagnosis).

The authors conclude that while the method shows strong performance in reconstruction and characterization, it is limited by the need for covariance estimation (which requires sufficient data density for smoothing) and the assumption of Gaussian latent distributions. Future work may explore non-Gaussian extensions and alternative smoothing techniques.