Tucker Decomposition Based on a Tensor Train of Coupled and Constrained CP Cores

Maxence Giraud, Vincent Itier, Rémy Boyer, Yassine Zniyed, André L. F. de Almeida

Abstract: Many real-life signal-based applications use the Tucker decomposition of a high dimensional/order tensor. A well-known problem with the Tucker model is that its number of entries increases exponentially with its order, a phenomenon known as the “curse of the dimensionality”. The Higher-Order Orthogonal Iteration (HOOI) and Higher-Order Singular Value Decomposition (HOSVD) are known as the gold standard for computing the range span of the factor matrices of a Tucker Decomposition but also suffer from the curse. In this letter, we propose a new methodology with a similar estimation accuracy as the HOSVD with non-exploding computational and storage costs. If the noise-free data follows a Tucker decomposition, the corresponding Tensor Train (TT) decomposition takes a remarkable specific structure. More precisely, we prove that for a $Q$ -order Tucker tensor, the corresponding TT decomposition is constituted by $Q-3$ 3-order TT-core tensors that follow a Constrained Canonical Polyadic Decomposition. Using this new formulation and the coupling property between neighboring TT-cores, we propose a JIRAFE-type scheme for the Tucker decomposition, called TRIDENT. Our numerical simulations show that the proposed method offers a drastically reduced complexity compared to the HOSVD and HOOI while outperforming the Fast Multilinear Projection (FMP) method in terms of estimation accuracy.