One-dimensional Tensor Network Recovery

TL;DR

This paper introduces efficient algorithms for recovering underlying graph structures or permutations in tensor ring and tensor train formats, demonstrating high accuracy and robustness in noiseless and noisy scenarios with theoretical guarantees and numerical validation.

Contribution

The paper presents novel algorithms with provable guarantees for graph and permutation recovery in tensor ring/train formats, with complexity $O(d\log d)$ and robustness to noise.

Findings

Algorithms recover graphs/permutations almost surely in noiseless cases.

Proven robustness of algorithms against observational noise.

Validated effectiveness through numerical experiments.

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.