Mathematics and Statistics 9(3): 394-410, 2021 DOI: 10.13189/ms.2021.090322 http://www.hrpub.org Tensor Multivariate Trace Inequalities and Their Applications Shih Yu Chang 1,* , Hsiao-Chun Wu 2 1 Department of Applied Data Science, San Jose State University, San Jose, CA 95192, USA 2 Division of Electrical & Computer Engineering, Louisiana State University, Baton Rouge, LA 70803 USA * Corresponding Author: shihyu.chang@sjsu.edu, hwu1@lsu.edu Received February 2, 2021; Revised March 28, 2021; Accepted April 18, 2021 Cite This Paper in the following Citation Styles (a): [1] Shih Yu Chang, Hsiao-ChunWu, ”Tensor Multivariate Trace Inequalities and Their Applications,” Mathematics and Statistics, Vol.9, No.3, pp. 394-410, 2021. DOI: 10.13189/ms.2021.090322 (b): Shih Yu Chang, Hsiao-ChunWu, (2021). Tensor Multivariate Trace Inequalities and Their Applications. Mathematics and Statistics, 9(3), 394-410. DOI: 10.13189/ms.2021.090322 Abstract In linear algebra, the trace of a square matrix is defined as the sum of elements on the main diagonal. The trace of a matrix is the sum of its eigenvalues (counted with multiplicities), and it is invariant under the change of basis. This charac- terization can be used to define the trace of a tensor in general. Trace inequalities are mathematical relations between different multivariate trace functionals involving linear operators. These relations are straightforward equalities if the involved linear operators commute, however, they can be difficult to prove when the non-commuting linear operators are involved. Given two Hermitian tensors H 1 and H 2 that do not commute. Does there exist a method to transform one of the two tensors such that they commute without completely destroying the structure of the original tensor? The spectral pinching method is a tool to resolve this problem. In this work, we will apply such spectral pinching method to prove several trace inequalities that extend the Araki–Lieb–Thirring (ALT) inequality, Golden–Thompson(GT) inequality and logarithmic trace inequality to arbitrary many tensors. Our approaches rely on complex interpolation theory as well as asymptotic spectral pinching, providing a transpar- ent mechanism to treat generic tensor multivariate trace inequalities. As an example application of our tensor extension of the Golden–Thompson inequality, we give the tail bound for the independent sum of tensors. Such bound will play a fundamental role in high-dimensional probability and statistical data analysis. Keywords Tensor, Multivariate, Trace, Golden–Thompson Inequality, Araki–Lieb–Thirring Inequality AMS 15A69,46B28,47H60,47A30 1 Introduction Trace inequalities are mathematical relations between different multivariate trace functionals involving linear operators. These relations are straightforward equalities if the involved linear operators commute, however, they can be difficult to prove when the non-commuting linear operators are involved [4]. One of the most important trace inequalities is the famous Golden-Thompson inequality [8]. For any two Hermitian matrices H 1 and H 2 , we have Tr exp(H 1 + H 2 ) ≤ Tr exp(H 1 ) exp(H 2 ). (1) It is easy to see that the Eq. (1) becomes an identity if two Hermitian matrices H 1 and H 2 are commute. The inequality in Eq. (1) has been generalized to several situations. For example, it has been demonstrated that it remains valid by replacing the trace with any unitarily invariant norm [14, 24]. The Golden-Thompson inequality has been applied to many various fields ranging from quantum information processing [16, 17], statistical physics [26, 29], and random matrix theory [1, 27].