Computational Mathematics and Computer Modeling with Applications (CMCMA)

Tensor LU and QR decompositions and their randomized algorithms

https://doi.org/10.52547/CMCMA.1.1.1
Volume 1, Issue 1 - Serial Number 1
June 2022
Pages 1-16
Computational Mathematics and Computer Modeling with Applications (CMCMA)

Document Type : Invited paper

Authors

Yuefeng Zhu 1
Yimin Wei 2

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

Abstract
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.

Keywords

LU decomposition
QR decomposition
rank-revealing algorithm
randomized algorithm
tensor T-product
low-rank approximation
  • Receive Date 10 December 2021
  • Revise Date 30 December 2021
  • Accept Date 31 December 2021

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