Quotient geometry of tensor ring decomposition

TL;DR

This paper develops a quotient geometric framework for tensor ring (TR) decomposition, addressing its intrinsic geometry and gauge invariance, and validates the approach through tensor completion experiments.

Contribution

It introduces the quotient geometry of TR decomposition under full-rank conditions, enhancing understanding of its intrinsic structure and extending to uniform TR decomposition.

Findings

Validated the geometric framework with tensor ring completion tasks

Established gauge invariance in the quotient geometry

Extended the framework to uniform TR decomposition

Abstract

Differential geometries derived from tensor decompositions have been extensively studied and provided the foundations for a variety of efficient numerical methods. Despite the practical success of the tensor ring (TR) decomposition, its intrinsic geometry remains less understood, primarily due to the underlying ring structure and the resulting nontrivial gauge invariance. We establish the quotient geometry of TR decomposition by imposing full-rank conditions on all unfolding matrices of the core tensors and capturing the gauge invariance. Additionally, the results can be extended to the uniform TR decomposition, where all core tensors are identical. Numerical experiments validate the developed geometries via tensor ring completion tasks.