Random Tensor Inequalities and Tail bounds for Bivariate Random Tensor Means, Part II

Abstract: This is Part II of our work about random tensor inequalities and tail bounds for bivariate random tensor means. After reviewing basic facts about random tensors, we first consider tail bounds with more general connection functions. Then, a general Lie-Trotter formula for tensors is derived and this formula is applied to establish tail bounds for bivariate random tensor means involving tensor logarithm. All random tensors studied in our Part I work are assumed as positive definite (PD) random tensors, which are invertible tensors. In this Part II work, we generalize our tail bounds for bivariate random tensor means from positive definite (PD) random tensors to positive semidefinite (PSD) random tensors by defining Random Tensor Topology (RTT) and developing the limitation method based on RTT. Finally, we apply our theory to establish tail bounds and L\"owner ordering relationships for bivariate random tensor means before and after two tensor data processing methods: data fusion and linear transform. %