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One-dimensional Tensor Network Recovery

Ziang Chen, Jianfeng Lu, Anru R. Zhang

TL;DR

This work tackles the problem of recovering the underlying one-dimensional tensor network topology (TR loop or TT path) from observed tensor entries by recovering the permutation that orders the indices. It introduces divide-and-conquer algorithms that test the correct order of small index subsets via rank or singular-value comparisons on down-sampled matricizations, achieving polynomial-time complexity with an overall sampling cost of $O(d log d)$. The authors prove almost-sure correct recovery in the noiseless setting under mild bond-dimension assumptions and establish high-probability robustness results under Gaussian observation noise, supplemented by extensive numerical experiments including a Potts-model case. The results enable reliable topology discovery in tensor networks, with implications for quantum physics, numerical analysis, and machine learning, especially when the underlying graph is unknown or dynamic.

Abstract

We study the recovery of the underlying graphs or permutations for tensors in the tensor ring or tensor train format. Our proposed algorithms compare the matricization ranks after down-sampling, whose complexity is $O(d\log d)$ for $d$-th order tensors. We prove that our algorithms can almost surely recover the correct graph or permutation when tensor entries can be observed without noise. We further establish the robustness of our algorithms against observational noise. The theoretical results are validated by numerical experiments.

One-dimensional Tensor Network Recovery

TL;DR

This work tackles the problem of recovering the underlying one-dimensional tensor network topology (TR loop or TT path) from observed tensor entries by recovering the permutation that orders the indices. It introduces divide-and-conquer algorithms that test the correct order of small index subsets via rank or singular-value comparisons on down-sampled matricizations, achieving polynomial-time complexity with an overall sampling cost of . The authors prove almost-sure correct recovery in the noiseless setting under mild bond-dimension assumptions and establish high-probability robustness results under Gaussian observation noise, supplemented by extensive numerical experiments including a Potts-model case. The results enable reliable topology discovery in tensor networks, with implications for quantum physics, numerical analysis, and machine learning, especially when the underlying graph is unknown or dynamic.

Abstract

We study the recovery of the underlying graphs or permutations for tensors in the tensor ring or tensor train format. Our proposed algorithms compare the matricization ranks after down-sampling, whose complexity is for -th order tensors. We prove that our algorithms can almost surely recover the correct graph or permutation when tensor entries can be observed without noise. We further establish the robustness of our algorithms against observational noise. The theoretical results are validated by numerical experiments.
Paper Structure (25 sections, 18 theorems, 73 equations, 10 figures, 4 algorithms)

This paper contains 25 sections, 18 theorems, 73 equations, 10 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Tensors
  4. One-dimensional tensor networks
  5. Tensor ring structure
  6. Tensor train structure
  7. Recovery of the underlying graph
  8. Tensor ring structure with permutation
  9. Tensor train structure with permutation
  10. Tasks and goals
  11. Proposed Algorithms
  12. Recovery of the underlying loop for tensor ring format
  13. Recovery of the underlying path for tensor train format
  14. Analysis of the Algorithms
  15. Recovery in the noiseless case
  16. ...and 10 more sections

Key Result

Proposition 3.2

Let $\tau,\tau'\in S_d$. $\tau'\in \mathcal{C}S_{\text{TR}}^d(\tau)$ if and only if for any $1\leq j_1<j_2<j_3<j_4\leq d$, $(\tau'(j_1),\tau'(j_2),\tau'(j_3),\tau'(j_4))$ is of the correct order with respect to $\tau$.

Figures (10)

  • Figure 1: Tensor ring format
  • Figure 2: Tensor train format
  • Figure 3: Tensor ring format with permutation
  • Figure 4: Tensor train format
  • Figure 5: Intuition of Algorithm \ref{['alg: 4index']}: when $(i_1,i_2,i_3,i_4)$ is of the correct order, the rank of $M_{(i_1,i_2),(i_3,i_4)}$ and $M_{(i_1,i_4),(i_2,i_3)}$ is at most $R^2$, while the rank of $M_{(i_1,i_3),(i_2,i_4)}$ is generically at least $R^4$. The reason is that the number of blue lines connecting red boxes is two in (A) and (B), but four in (C), where red boxes group the column indices in the matricization.
  • ...and 5 more figures

Theorems & Definitions (37)

  • Definition 3.1
  • Proposition 3.2
  • Remark 3.3
  • Definition 3.4
  • Remark 3.5
  • Definition 3.6
  • Proposition 3.7
  • Definition 3.8
  • Theorem 4.2
  • Corollary 4.3
  • ...and 27 more