A TT-Based Hierarchical Framework for Decomposing High-Order Tensors

In the context of big data, high-order tensor decompositions have to face a new challenge in terms of storage and computational costs. The tensor train (TT) decomposition provides a very useful graph-based model reduction, whose storage cost grows linearly with the tensor order D. The computation of the TT-core tensors and TT-ranks can be done in a stable sequential (i.e., non-iterative) way thanks to the popular TT-SVD algorithm. In this paper, we exploit the ideas developed for the hierarchical/tree Tucker decomposition in the context of the TT decomposition. Specifically, a new efficient estimation scheme, called TT-HSVD for Tensor-Train Hierarchical SVD, is proposed as a solution to compute the TT decomposition of a high-order tensor. The new algorithm simultaneously delivers the TT-core tensors and their TT-ranks in a hierarchical way. It is a stable (i.e., non-iterative) and computationally more efficient algorithm than the TT-SVD one, which is very important when dealing with large-scale data. The TT-HSVD algorithm uses a new reshaping strategy and a tailored partial SVD, which allows to deal with smaller matrices compared to those of the TT-SVD. In addition, TT-HSVD suits well for a parallel processing architecture. An algebraic analysis of the two algorithms is carried out, showing that TT-SVD and TT-HSVD compute the same TT-ranks and TT-core tensors up to specific bases. Simulation results for different tensor orders and dimensions corroborate the effectiveness of the proposed algorithm.