Tensor Ring Decomposition Achieves Exact Solutions with a Deterministic, Finite-Step Algorithm

Tensor networks represent a powerful tool for handling the massive datasets common in modern science, yet efficiently decomposing these networks remains a significant challenge. Han Chen, Sitan Chen, and Anru R. Zhang from Duke University now present a provably efficient method for tensor ring decomposition, resolving a long-standing question about the existence of a reliable, step-by-step procedure. Their innovative approach utilises blockwise simultaneous diagonalization to extract key information from limited tensor observations, offering both a deeper understanding of the underlying mathematics and a practical advantage in computational speed. This work extends to symmetric tensor ring decomposition, simplifying the process and opening doors to applications in areas such as physics-based modelling and the analysis of complex data, while a robust recovery scheme improves accuracy even when dealing with noisy data, ultimately advancing the foundations of scalable tensor network computation.

To extend the applicability of tensor ring decomposition, the research focuses on the symmetric tensor ring setting, where parameter complexity is significantly reduced and applications arise naturally in physics-based modelling and exchangeable data analysis. To improve the robustness of the decomposition in the presence of noisy data, the team developed a recovery scheme that combines their initialisation method with an alternating least squares approach, achieving faster convergence and improved accuracy compared to existing techniques.

BLOSTR Algorithm Performance with Noise and Rank

Scientists have comprehensively evaluated the performance of BLOSTR, an algorithm designed for tensor decomposition with a specified rank. Extensive numerical experiments, varying the tensor rank, noise levels, and initialisation methods, demonstrate that BLOSTR consistently outperforms a randomly initialised algorithm, particularly in the presence of noise. Increasing the number of iterations generally improves performance for both algorithms, but the improvement is more pronounced for BLOSTR. As the rank increases, the reconstruction error tends to increase, as expected, confirming that BLOSTR is a robust and reliable algorithm for tensor decomposition, offering significant advantages over standard approaches.

Deterministic Tensor Core Recovery via BLOSTR

Scientists have achieved a breakthrough in tensor ring decomposition, developing the first deterministic, finite-step algorithm to exactly recover tensor cores from observed data. This work addresses a long-standing question regarding the existence of such a procedure and introduces a method leveraging blockwise simultaneous diagonalization to achieve this goal. The team demonstrates the ability to extract tensor cores from a limited number of tensor observations, providing both algebraic insight and practical efficiency for tensor decomposition. Experiments reveal that BLOSTR successfully decomposes tensors into their constituent cores, even with limited data, by identifying a representative set of tensor cores. For order-2 tensors, which are equivalent to matrices, the algorithm directly corresponds to a singular value decomposition, providing a clear connection to established techniques.

Deterministic Tensor Decomposition via Block Diagonalization

This work presents a novel, deterministic algorithm for decomposing tensors using tensor ring decomposition. Addressing a long-standing open question in the field, the method efficiently recovers the core components of a tensor from a limited set of observations. The approach utilizes blockwise simultaneous diagonalization, providing both a deeper algebraic understanding of tensor decomposition and practical improvements in computational efficiency. Importantly, the researchers extended this method to the symmetric tensor ring setting, which simplifies the process and is particularly relevant to modeling physical systems and analyzing data where order doesn’t matter. The researchers demonstrated the versatility of their algorithm by applying it to problems in other areas, including matrix product state tomography, a technique used in quantum information, and the calculation of pushforward distributions, a foundational concept in mathematics. These achievements advance the algorithmic foundations of tensor ring decomposition and open new possibilities for scalable computation with tensor networks.