T-square tensors—Part I: inequalities

Abstract

The T-square operation between two third-order tensors was invented around 2011 and it arises from many applications, such as signal processing, image feature extraction, machine learning, computer vision, and the multi-view clustering problem. Although there are many pioneer works about T-square tensors, there are no works dedicated to inequalities associated with T-square tensors. In this work, we first attempt to build inequalities at the following aspects: (1) trace function nondecreasing/convexity; (2) Golden–Thompson inequality for T-square tensors; (3) Jensen’s T-square inequality; (4) Klein’s T-square inequality. All these inequalities are related to generalize celebrated Lieb’s concavity theorem from matrices to T-square tensors. This new version of Lieb’s concavity theorem under T-square tensor will be used to determine the tail bound for the maximum eigenvalue induced by independent sums of random Hermitian T-square, which is the key tool to derive various new tail bounds for random T-square tensors. Besides, Qi and Zhang (T-quadratic forms and spectral analysis of t-symmetric tensors, 2021) introduces a new concept, named eigentuple, about T-square tensors and they apply this concept to study nonnegative (positive) definite properties of T-square tensors. The final main contribution of this work is to develop the Courant–Fischer Theorem with respect to eigentuples, and this theorem helps us to understand the relationship between the minimum eigentuple and the maximum eigentuple. The main content of this paper is Part I of a serious task about T-square tensors. The Part II of this work will utilize these new inequalities and Courant–Fischer Theorem under T-square tensors to derive tail bounds of the extreme eigenvalue and the maximum eigentuple for sums of random T-square tensors, e.g., T-square tensor Chernoff and T-square tensor Bernstein bounds.