STOCHASTIC CONDITIONING OF TENSOR FUNCTIONS BASED ON THE TENSOR-TENSOR PRODUCT

摘要

The conditioning of matrix functions is one of the fundamental topics in linear algebra. In this paper, we extend matrix case to third order tensor functions based on the tensor-tensor product. We first give the deterministic perturbation bounds for the tensor Moore-Penrose inverse based on the tensor -tensor product and generalized them to T-Total Least Squares problem. Then, we discuss the bound of the stochastic perturbations of third order tensors. We investigate the stochastic conditioning problem for general tensor function if random noises are input. We define the Frechet derivative of the generalized tensor function and give the upper bound of stochastic condition number and compare it with the deterministic condition number in the first order estimation. The stochastic conditioning will be better than the deterministic conditioning. Finally, we present a numerical example of the tensor least squares problem to show the effectiveness of our stochastic error estimate.