Enhanced Solutions for the Block-Term Decomposition in Rank-$(L_{r},L_{r},1)$ Terms

The block-term decompositions (BTD) represent tensors as a linear combination of low multilinear rank terms and can be explicitly related to the Canonical Polyadic decomposition (CPD). In this paper, we introduce the SECSI-BTD framework, which exploits the connection between two decompositions to estimate the block-terms of the rank-
$(L_{r},L_{r},1)$
BTD. The proposed SECSI-BTD algorithm includes the initial calculation of the factor estimates using the SEmi-algebraic framework for approximate Canonical polyadic decompositions via SImultaneous Matrix Diagonalizations (SECSI), followed by clustering and refinement procedures that return the appropriate rank-
$(L_{r},L_{r},1)$
BTD terms. Moreover, we introduce a new approach to estimate the multilinear rank structure of the tensor based on the HOSVD and
$k$
-means clustering. Since the proposed SECSI-BTD algorithm does not require a known rank structure but can still take advantage of the known ranks when available, it is more flexible than the existing techniques in the literature. Additionally, our algorithm does not require multiple initializations, and the simulation results show that it provides more accurate results and a better convergence behavior for an extensive range of SNRs.