Tensor Multivariate Trace Inequalities and their Applications

TL;DR

This paper extends key trace inequalities to multiple tensors using complex interpolation and spectral pinching, enabling new bounds in high-dimensional probability and data analysis.

Contribution

It introduces tensor multivariate trace inequalities extending classical results, with novel proof techniques and applications to tail bounds in high-dimensional settings.

Findings

Extended ALT, GT, and logarithmic trace inequalities to tensors

Provided tail bounds for sums of tensors

Established a framework for tensor inequalities in high-dimensional analysis

Abstract

We 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.