Spectral Bounds for Tensors Derived from Trace Functionals and Wasserstein Distance in Tensor Spaces
Provides theoretical foundations for measuring distances and bounding eigenvalues in tensor spaces, relevant for researchers in tensor analysis and optimization.
This paper introduces a trace-based metric (Bures-Wasserstein distance) on the space of positive semi-definite tensors, derives trace-based eigenvalue bounds, and analyzes their dependence on the PSD condition, with complexity analysis. No concrete numerical results are reported.
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.