Paper Info
Title
A TT-BASED HIERARCHICAL FRAMEWORK FOR DECOMPOSING HIGH-ORDER TENSORS
Abstract
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., noniterative) 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 (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., noniterative) and computationally more efficient algorithm than TT-SVD, 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 us to deal with smaller matrices compared to those of the TT-SVD. In addition, TT-HSVD is well suited 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.
Year
DOI
Venue
2020
10.1137/18M1229973
SIAM JOURNAL ON SCIENTIFIC COMPUTING
Keywords
DocType
Volume
tensor train,hierarchical SVD,dimensionality reduction,tensor graph
Journal
42
Issue
ISSN
Citations 
2
1064-8275
1
PageRank 
References 
Authors
0.39
0
4
Name
Order
Citations
PageRank
Yassine Zniyed132.22
Rémy Boyer230138.10
André L. F. de Almeida337148.73
GéRard Favier451446.41