Tensor Multivariate Trace Inequalities and Their Applications

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 characterization 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 H1 and H2 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 transparent 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.