Tensor decomposition with a pre-defined structure

Implementations for tensor decomposition subject to structural constraints with linear dependencies among latent components.

The presented MATLAB package provides functions for Block-Term Tensor Decomposition (BTD) [1] and Parallel Profiles with Linear Dependencies (PARALIND) [2].

BLOCK-TERM DECOMPOSITION (BTD)

Component matrices are estimated via the least squares algorithm. For each mode, users can specify constraints on component matrices, including orthogonality, non-negativity, or unimodality. Additionally, the number of latent components can be customized independently within each block.

PARALLEL PROFILES WITH LINEAR DEPENDENCIES (PARALIND)

Building upon the original PARALIND MATLAB script [3], this implementation extends functionality to support dependency matrices across all three modes simultaneously. Users may apply various constraints to both component and dependency matrices, such as non-negativity, unimodality or orthogonality (the last two for component matrices only). Additionally, either component or dependency matrices can be fixed at pre-defined values during estimation.

Three algorithms are available for estimating component and dependency matrices:

1. PARAFAC-based approach: A PARAFAC model [5,6] is first fitted to the tensor data, then each estimated PARAFAC factor matrix is decomposed into component and dependency matrices via generalized matrix decomposition.

2. Tucker-based approach: A Tucker model [7] is applied to obtain component matrices. Dependency matrices are subsequently estimated by performing PARAFAC decomposition on the core tensor.

3. Alternating Least Squares (ALS): Component and dependency matrices are jointly estimated from the tensor data using an iterative ALS procedure.

Refrences:

[1] L. De Lathauwer, Decompositions of a higher-order tensor in block terms - Part II: Definitions and uniqueness, SIAM Journal on Matrix Analysis and Applications 30 (3) (2008) 1033–1066. doi:10.1137/070690729.

[2] R. Bro, R. A. Harshman, N. D. Sidiropoulos, M. E. Lundy, Modeling multi-way data with linearly dependent loadings, Journal of Chemometrics: A Journal of the Chemometrics Society 23 (7-8) (2009) 324–340.

The original MATLAB code for PARALIND (which served as an inspiration) can be downloaded from

[3] https://ucphchemometrics.com/186-2/algorithms/

[4] R. A. Harshman, Foundations of the PARAFAC procedure: Models and conditions for an “explanatory” multimodal factor analysis, UCLA Working Papers in Phonetics 16 (1970) 1–84.

[5] J. D. Carroll, J.-J. Chang, Analysis of individual differences in multidimensional scaling via an N-way generalization of “Eckart-Young” decomposition, Psychometrika 35 (3) (1970) 283–319. doi:https://doi.org/10.1007/BF02310791.

[6] L. R. Tucker, Some mathematical notes on three-mode factor analysis, Psychometrika 31 (3) (1966) 279–311. doi:https://doi.org/10.1007/BF02289464.

These MATLAB scripts were developed within the EU NextGenerationEU through the Recovery and Resilience Plan for Slovakia under the project No. 09I03-03-V04-00205.