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A Provably Efficient Method for Tensor Ring Decomposition and Its Applications

Han Chen, Sitan Chen, Anru R. Zhang

TL;DR

The paper resolves the open question of whether a finite-step, deterministic algorithm exists for exact tensor ring (TR) decomposition by introducing BLOSTR, a Blockwise Simultaneous Diagonalization method that recovers TR cores from sparse observations. It proves finite-step exact recovery under identifiable conditions and shows sampling optimality with $O((n_1+ o n_d) r^2)$ observations. The authors extend the framework to symmetric TR, and develop a robust BLOSTR-ALS hybrid to handle perturbations, missing data, and noise, with strong empirical performance. They also connect TR decomposition to matrix product state tomography and moment-tensor learning, highlighting broad applicability in quantum information and high-dimensional statistics, and outline future directions for heterogeneous ranks and explicit noisy guarantees.

Abstract

We present the first deterministic, finite-step algorithm for exact tensor ring (TR) decomposition, addressing an open question about the existence of such procedures. Our method leverages blockwise simultaneous diagonalization to recover TR-cores from a limited number of tensor observations, providing both algebraic insight and practical efficiency. We extend the approach to the symmetric TR setting, where parameter complexity is significantly reduced and applications arise naturally in physics-based modeling and exchangeable data analysis. To handle noisy observations, we develop a robust recovery scheme that couples our initialization with alternating least squares, achieving faster convergence and improved accuracy compared to classic methods. As applications, we obtain new algorithms for questions in other domains where tensor ring decomposition is a key primitive, namely matrix product state tomography in quantum information, and provable learning of pushforward distributions in the foundations of machine learning. These contributions advance the algorithmic foundations of TR decomposition and open new opportunities for scalable tensor network computation.

A Provably Efficient Method for Tensor Ring Decomposition and Its Applications

TL;DR

The paper resolves the open question of whether a finite-step, deterministic algorithm exists for exact tensor ring (TR) decomposition by introducing BLOSTR, a Blockwise Simultaneous Diagonalization method that recovers TR cores from sparse observations. It proves finite-step exact recovery under identifiable conditions and shows sampling optimality with observations. The authors extend the framework to symmetric TR, and develop a robust BLOSTR-ALS hybrid to handle perturbations, missing data, and noise, with strong empirical performance. They also connect TR decomposition to matrix product state tomography and moment-tensor learning, highlighting broad applicability in quantum information and high-dimensional statistics, and outline future directions for heterogeneous ranks and explicit noisy guarantees.

Abstract

We present the first deterministic, finite-step algorithm for exact tensor ring (TR) decomposition, addressing an open question about the existence of such procedures. Our method leverages blockwise simultaneous diagonalization to recover TR-cores from a limited number of tensor observations, providing both algebraic insight and practical efficiency. We extend the approach to the symmetric TR setting, where parameter complexity is significantly reduced and applications arise naturally in physics-based modeling and exchangeable data analysis. To handle noisy observations, we develop a robust recovery scheme that couples our initialization with alternating least squares, achieving faster convergence and improved accuracy compared to classic methods. As applications, we obtain new algorithms for questions in other domains where tensor ring decomposition is a key primitive, namely matrix product state tomography in quantum information, and provable learning of pushforward distributions in the foundations of machine learning. These contributions advance the algorithmic foundations of TR decomposition and open new opportunities for scalable tensor network computation.

Paper Structure

This paper contains 26 sections, 13 theorems, 82 equations, 9 figures, 1 table, 6 algorithms.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Order-2 TR Decomposition.
  4. Ranks of TR Decomposition.
  5. Finite-Step TR Decomposition via Blockwise Simultaneous Diagonalization
  6. Step 1 (two spectral probes).
  7. Step 2 (block identification / gauge fixing).
  8. Step 3 (recover $\mathbfcal{Q}_1$).
  9. Step 4 (circular permutation and remaining cores).
  10. BLOSTR for Symmetric Tensor-Ring Decomposition
  11. TR Decomposition in Presence of Perturbation
  12. Numerical Studies
  13. Connections to Matrix Product States and Moment Tensors
  14. Matrix Product States
  15. Learning Polynomial Transformations
  16. ...and 11 more sections

Key Result

Lemma 1

Suppose $\mathbfcal{T}\in\mathbb{C}^{n_1\times n_2\times\cdots\times n_d}$ satisfies eq:order4TR, where $d\geq 3$, $r\geq 2$, and $n_k \geq r^2$ for all $k\in[d]$. Assume that the elements of $\mathbfcal{Q}_k\in\mathbb{C}^{n_k\times r\times r}$ are randomly drawn from a measure $\mu$ that is absolut where $\mathbf{K}_j\in\mathop{\mathrm{GL}}\nolimits(r,\mathbb{C})$, $j\in[r]$. Equivalently, restri

Figures (9)

  • Figure 1: Illustration of $\Pi_{2,3}$ operated on a 6-by-6 matrix $\mathbf{X}$. Elements in the same sub-matrices are labeled with same colors.
  • Figure 2: Illustrative example of a possible choice of $\Delta$ in a three-way tensor. Each small cell represents an entry. We set $n_1=n_2=n_3=10$, $r=2$, and all $\Gamma_{\cdot}$'s are $\{1,2,3,4\}$.
  • Figure 3: Average relative error (with standard deviation in log scale) of BLOSTR (Algorithm \ref{['alg:noise']}) and randomly initialized ALS over varying numbers of iterations. Full: access to full entries of tensor $\mathbfcal{T}$ at the stage of ALS; $\Delta$: ALS with entries limited to $\Delta$ in \ref{['eq:Delta']}. The entries of $\{\mathbfcal{Q}_k\}_{k\in[d]}$ are independently drawn from $\mathcal{N}(0, 10^2)$, and the variance of the added Gaussian noise is set to 1. The tensor $\mathbfcal{T}$ has dimension $30^{\times 3}$ in the top row and $30^{\times 4}$ in the bottom row.
  • Figure 4: Average relative error (with standard deviation in log scale) of BLOSTR (Algorithm \ref{['alg:noise']}) and randomly initialized ALS over increasing noise scales with 3 iterations. Full: access to full entries of tensor $\mathbfcal{T}$ at the stage of ALS; $\Delta$: ALS with entries limited to $\Delta$ in \ref{['eq:Delta']}. The entries of $\{\mathbfcal{Q}_k\}_{k\in[d]}$ are independently drawn from $\mathcal{N}(0, 10^2)$.The tensor $\mathbfcal{T}$ has dimension $30^{\times 3}$ in the top row and $30^{\times 4}$ in the bottom row.
  • Figure 5: Proportion of successful recoveries (relative error below $10^{-5}$) within at most 10 iterations, averaged over 100 independent trials, using BLOSTR (Algorithm \ref{['alg:noise']}) under varying noise scales $\sigma_n$. Each tensor core $\{\mathbfcal{Q}_k\}_{k\in[d]}$ has entries independently drawn from $\mathcal{N}(0, 10^2)$. The underlying tensor $\mathbfcal{T}$ has dimension $30^{\times 3}$ in the top row and $30^{\times 4}$ in the bottom row.
  • ...and 4 more figures

Theorems & Definitions (30)

  • Lemma 1
  • Lemma 2: zhao2016tensor
  • Theorem 1: BLOSTR achieves exact tensor ring decomposition in finite steps
  • Remark 1
  • Remark 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Theorem 2: Blockwise Simultaneous Diagonalization Method for Symmetric Tensor Ring Decomposition
  • Remark 3: TR Decomposition in Presence of Perturbation
  • ...and 20 more