A unifying framework for modelling non-negative bi-linear, tri-linear and “in-between” data in chemometrics. Part I: Theoretical framework and concepts

Abstract

In chemometrics, extracting chemically meaningful information from multi-way analytical data is often challenged by deviations from ideal tri-linear structure of the chemical information. This work introduces a novel modeling approach based on (1, $L_r$, $L_r$) block term decompositions, which flexibly bridges the gap between bi-linear and tri-linear models. The method builds upon the MCR-tri-linearity framework and leverages uniqueness conditions established by De Lathauwer to ensure interpretable factor solutions under practical conditions. A rank-constrained alternating optimization algorithm is proposed to adaptively determine the number of principal components needed for reconstructing varying-mode factors, based on a user-defined reconstruction error tolerance. This adaptive decomposition balances the essential uniqueness of tri-linear models with the flexibility of bi-linear approaches, addressing limitations in both. Simulated data with controlled component ranks demonstrate the method's ability to recover ground-truth factors more accurately than classical tri-linear models, while reducing ambiguity compared to bi-linear models. The results confirm that the proposed approach provides an effective framework for analyzing multi-way chemical data with partial or full deviations from tri-linearity, making it a promising tool for a wide range of chemometric applications.