Spectral Bounds for Tensors Derived from Trace Functionals and Wasserstein Distance in Tensor Spaces

TL;DR

This paper develops a trace-based metric and Bures-Wasserstein distance for PSD tensors, deriving eigenvalue bounds and analyzing their behavior under PSD relaxation, supported by complexity analysis and examples.

Contribution

It introduces a novel trace-based metric and eigenvalue bounds for PSD tensors, linking algebraic and geometric properties in tensor spaces.

Findings

Established a trace-based eigenvalue bound for PSD tensors.

Analyzed the dependence of bounds on the PSD condition.

Provided illustrative examples and complexity analysis.

Abstract

This article introduces a trace-based metric on the space of positive semi-definite (PSD) tensors, offering a geometric perspective that connects their algebraic structure to their intrinsic geometric properties. It defines the Bures-Wasserstein distance on tensor spaces, establishing clear measurements between tensors. Moreover, the study derives trace-based eigenvalue bounds for PSD tensors and analyzes how these bounds depend on the PSD condition. The behavior of these bounds is further explored when the PSD requirement is relaxed, with illustrative examples provided to support the theoretical findings. In addition, a detailed complexity analysis is carried out for the methods proposed in this study.