Core consistency diagnosis for Block Term Decomposition in rank $(L_r, L_r, 1)$

Abstract

Determining the underlying number of components $R$ in tensor decompositions is challenging. Diverse techniques exist for various decompositions, notably the core consistency diagnostic (CORCONDIA) for Canonical Polyadic Decomposition (CPD). Here, we propose a model that intuitively adapts CORCONDIA for rank estimation in Block Term Decomposition (BTD) of rank $(L_r, L_r, 1)$: BTDCORCONDIA. Our metric was tested on simulated and real-world tensor data, including assessments of its sensitivity to noise and the indeterminacy of BTD $(L_r, L_r, 1)$. We found that selecting appropriate $R$ and $L_r$ led to core consistency reaching or close to 100%, and BTDCORCONDIA is efficient when the tensor has significantly more elements than the core. Our results confirm that CORCONDIA can be extended to BTD $(L_r, L_r, 1)$, and the resulting metric can assist in the process of determining the number of components in this tensor factorisation.