On The Accelerations Of TT Decomposition

In the context of big data,tensors are widely applied and how to represent tensors by a small number of parameters becomes a hot topic.Due to the curse of dimensional-ity,high-order tensor decompositions have to face a new challenge in terms of storage and computational costs.The Tensor Train(TT)decomposition provides an efficient representation of tensors and can be done by a stable algorithm.In this dissertation,we mainly study the TT decomposition algorithm and its accelerations.The main contributions of this dissertation are as follows:(1)The definition of ?-- rearrangement is given,and it is theoretically proved that basic linear algebra operations on the tensor ?--rearranged is still valid in the TT format.If its result is a tensor,we just need to do the inverse operation of the?-- rearrangement on it.Based on this conclusion,the influence of different?-- rearrangements of tensors on the computational costs of TT decomposition is discussed.It is found that in the case of full-rank,the computational costs of TT decomposition is lower while the tensor is rearranged according to a certain law.The numerical results show that the rearrangement of tensors according to this law can accelerate TT decomposition.(2)Considering the characteristics of the SVD in the process of TT decomposition in the case of full-rank and combining with the hierarchical structure,a new decom-position strategy is proposed: TT decomposition is carried out by unfolding the tensor from the middle of the total dimension,and then continuing the decomposi-tion on the two retained matrices in different directions,from right to left and from left to right respectively.This strategy can effectively reduce the computational costs,and is also suitable for parallel processing.Both theoretical analysis and numerical examples demonstrate that the new strategy is more efficient.