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Pré-Publication, Document De Travail Année : 2023

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

Résumé

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 paper, 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 TTcores, 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.
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Dates et versions

hal-04112156 , version 1 (31-05-2023)

Identifiants

  • HAL Id : hal-04112156 , version 1

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Maxence Giraud, Vincent Itier, Remy Boyer, Yassine Zniyed, André de Almeida. Tucker Decomposition Based on a Tensor Train of Coupled and Constrained CP Cores. 2023. ⟨hal-04112156⟩
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