Tensor LU and QR decompositions and their randomized algorithms

Document Type : Invited paper


1 School of Mathematical Sciences, Fudan University, Shanghai, P.R. China

2 School of Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied Mathematics, Fudan University, Shanghai, PR China


In this paper, we propose two decompositions extended from matrices to tensors, including LU and QR decompositions with their rank-revealing  and  randomized variations. We give the growth order analysis of error of the tensor QR (t-QR) and tensor LU (t-LU) decompositions. Growth order of error and running time are shown by numerical  examples. We test our methods by compressing and analyzing the image-based data, showing that the performance of tensor randomized QR decomposition is better than the tensor randomized SVD (t-rSVD) in terms of the accuracy, running time and memory.


  • Receive Date: 10 December 2021
  • Revise Date: 30 December 2021
  • Accept Date: 31 December 2021
  • First Publish Date: 01 January 2022