Tensor LU and QR decompositions and their randomized algorithms

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.