A Provably Efficient Method for Tensor Ring Decomposition and Its Applications

Abstract

We present the first deterministic, finite-step algorithm for exact tensor ring (TR) decomposition. Our method leverages blockwise simultaneous diagonalization to recover TR cores from a limited number of tensor observations, under a dimension condition requiring each mode size to be at least quadratic in the TR rank and under a genericity assumption on the cores, thereby 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 classical 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.