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Quotient geometry of tensor ring decomposition

Bin Gao, Renfeng Peng, Ya-xiang Yuan

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.

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 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.
Paper Structure (24 sections, 10 theorems, 50 equations, 8 figures, 1 table, 4 algorithms)

This paper contains 24 sections, 10 theorems, 50 equations, 8 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related work and motivation
  3. Contributions
  4. Organization
  5. Preliminaries
  6. Notation for Riemannian geometry
  7. Notation for tensor operations
  8. Tensor ring decomposition
  9. Quotient geometry of tensor ring decomposition
  10. Gauge invariance
  11. Quotient geometry
  12. Vertical and horizontal spaces
  13. Discussion: connection to tensor train decomposition
  14. Higher compressibility of TR
  15. Different quotient gemetries
  16. ...and 9 more sections

Key Result

Theorem 3.1

Let $\mathcal{X}=\llbracket\mathcal{U}_1,\mathcal{U}_{2},\dots,\mathcal{U}_d\rrbracket$ satisfy $\mathrm{rank}((\mathcal{U}_k)_{(2)})=r_kr_{k+1}$ and $r_kr_{k+1}\leq n_k$ for all $k\in[d]$ and $d\geq 3$. If $\mathcal{X}$ can also be expressed by $\llbracket\mathcal{V}_1,\mathcal{V}_2,\dots,\mathcal{ where $\mathbf{A}_{d+1}:=\mathbf{A}_1$. Moreover, $\vec{\mathbf{A}}:=(\mathbf{A}_1,\mathbf{A}_2,\do

Figures (8)

  • Figure 1: Illustration of tensor ring decomposition, tensor train decomposition, and uniform tensor ring decomposition of a tensor via tensor networks.
  • Figure 1: Gauge invariance of tensor ring decomposition.
  • Figure 1: Commutative diagram of smooth functions on the total space ${\overline{\mathcal{M}}}_\mathbf{r}^*$ and quotient manifold $\mathcal{M}_\mathbf{r}^*$.
  • Figure 1: Recovery performance of TR-based and uniform TR-based methods in tensor ring completion under different geometries.
  • Figure 2: Commutative diagram of injective TR tensors
  • ...and 3 more figures

Theorems & Definitions (20)

  • Theorem 3.1: fundamental theorem of MPS
  • Definition 3.2: injective TR tensor
  • Lemma 3.3
  • Proof 1
  • Proposition 3.4
  • Proof 2
  • Remark 3.5: Necessity of the projective group
  • Proposition 3.6
  • Proof 3
  • Proposition 3.7
  • ...and 10 more