Quotient geometry of tensor ring decomposition

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.

💡 Research Summary

This paper develops a rigorous quotient‑manifold framework for the Tensor Ring (TR) decomposition, addressing the long‑standing gap in understanding its intrinsic geometry. The authors begin by observing that TR, unlike the Tensor Train (TT), possesses a cyclic structure that introduces a non‑trivial gauge invariance: inserting any invertible matrix Aₖ and its inverse between adjacent core tensors leaves the reconstructed full tensor unchanged. To obtain a well‑posed parametrization, they impose a full‑rank condition on each core’s mode‑2 unfolding, i.e., rank((Uₖ)^{(2)}) = rₖ·rₖ₊₁ with rₖ·rₖ₊₁ ≤ nₖ. Under this injectivity assumption, Theorem 3.1 (the fundamental theorem of Matrix Product States) guarantees that any two TR representations of the same tensor differ only by a sequence of invertible transformations (A₁,…,A_d) ∈ GL(r₁)×…×GL(r_d), unique up to a global scalar factor.

Consequently, the authors define a new parameter space M*₍r₎ consisting of all core tensors satisfying the full‑rank condition and introduce the projective linear group PGL(r) = (GL(r₁)×…×GL(r_d))/ℝ* to capture both the per‑core gauge freedom and the overall scaling ambiguity. They prove that the group action of PGL(r) on M*₍r₎ is free and proper, which implies that the quotient set M*₍r₎/PGL(r) is a smooth manifold—a quotient manifold that faithfully represents the space of injective TR tensors.

The paper then constructs the differential‑geometric tools required for optimization on this manifold. The vertical space VₓM is identified as the tangent to the group orbit and is explicitly expressed via infinitesimal generators Ωₖ ∈ 𝔤𝔩(rₖ). Because of the ring topology, the horizontal space HₓM (the orthogonal complement of VₓM under a chosen Riemannian metric) involves coupling conditions between adjacent cores, a notable departure from the TT case where each core can be treated independently. Proposition 3.10 provides a concrete characterization of HₓM, and the authors derive closed‑form projection operators onto VₓM and HₓM. With these, they define a retraction, vector transport, and the Riemannian gradient, enabling the implementation of standard Riemannian optimization algorithms (gradient descent, conjugate gradient, trust‑region) on the TR quotient manifold.

A significant extension is presented for the uniform Tensor Ring (Uniform‑TR) where all cores are identical. Although the quotient geometry of Uniform‑TR was previously noted in the literature, the authors supply explicit parametrizations of the vertical and horizontal spaces specific to this symmetric setting, dramatically reducing the number of free parameters and simplifying the projection formulas.

Experimental validation focuses on tensor completion tasks. Synthetic d‑order tensors (d=5, each mode size 20) are sampled at various observation ratios (10–30%). The authors compare three algorithms: (i) a TR‑based Riemannian gradient method on the newly derived manifold, (ii) a Uniform‑TR counterpart, and (iii) a baseline TT method. Results show that Uniform‑TR requires substantially fewer samples to achieve successful recovery and converges faster than the general TR approach, confirming that the reduced parameterization and the removal of redundant gauge degrees of freedom translate into practical performance gains.

In summary, the paper makes the following contributions: (1) introduces a full‑rank injectivity condition that resolves non‑uniqueness in TR representations; (2) constructs the quotient manifold M*₍r₎/PGL(r) with explicit vertical/horizontal decompositions and projection operators; (3) extends the framework to the symmetric Uniform‑TR case; (4) demonstrates how these geometric insights lead to efficient Riemannian optimization algorithms; and (5) validates the theory through comprehensive numerical experiments. The work bridges the gap between TR’s powerful expressive capacity and a solid geometric foundation, opening avenues for further research on low‑rank tensor models with cyclic structures, adaptive rank selection, and connections to quantum many‑body physics.