A New Tucker Model for Three-Way Functional Data: FTucker-3

Abstract

This paper introduces FTucker-3, a basis-preserving Tucker framework for three-way functional data in which each tensor cell is a function. Functional observations are represented in a common finite basis, organized as a fourth-order coefficient tensor, and reduced only along the three structural modes while preserving the selected functional coefficient mode. A preliminary theoretical characterization establishes an exact functional–coefficient error decomposition, identifiability of the structural loading subspaces under full-row-rank core unfoldings, and perturbation control for the conditional SVD updates. Numerical evaluation combined a five-scenario Monte Carlo study with 1500 fitted models, a controlled periodic experiment, a real NASA POWER application, a Tucker-4 benchmark, and a scalability analysis. In the controlled experiment, FTucker-3 explained 96.71% of coefficient variability and 97.14% of functional variability. For NASA POWER, leakage-free cross-validation selected ranks (4,5,1), yielding 94.67% coefficient-domain and 69.45% functional explained variability across 240 daily trajectories. Under functional and structural misspecification, dominant loading-subspace similarities remained above 0.999 at 5 dB. Tucker-4 compression reduced storage but also reduced functional reconstruction accuracy, whereas the full-rank fourth-mode case was numerically equivalent to FTucker-3. These results support FTucker-3 as an interpretable framework for three-way arrays of functions.

The authors' abstract, as published at the source. Mathematics, 2026 · DOI ↗