Quotient geometry of tensor ring 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 and immersed-submanifold structure of TR decomposition by imposing full-rank conditions on all unfolding matrices of the core tensors and capturing the gauge invariance. The intrinsic ring structure of TR leads to an analysis that is substantially different from other tensor formats. Additionally, for the uniform TR decomposition, where all core tensors are identical and the manifold structure is known, we derive explicit parameterizations for the vertical and horizontal spaces, which enable Riemannian optimization. Numerical experiments validate the developed geometries via tensor ring completion tasks.

Numerical Analysis (math.NA); Optimization and Control (math.OC); Quantum Physics (quant-ph)

arXiv:2601.21874 [math.NA]

(or arXiv:2601.21874v2 [math.NA] for this version)

https://doi.org/10.48550/arXiv.2601.21874 Focus to learn more arXiv-issued DOI via DataCite

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