A Provably Efficient Method for Tensor Ring Decomposition and Its Applications

TL;DR

This paper introduces the first deterministic, finite-step algorithm for exact tensor ring decomposition, with extensions to symmetric cases and noisy data, enabling efficient applications in physics, quantum information, and machine learning.

Contribution

It provides the first provably correct finite-step algorithm for tensor ring decomposition, including robust methods for noisy data and extensions to symmetric tensor rings.

Findings

Algorithm achieves exact TR decomposition with limited observations.

Extension to symmetric TR reduces parameter complexity.

Robust recovery scheme improves convergence and accuracy.

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.