Linear algebra at exponential scale via tensor network dimension reduction

2026-06-13

AI Summary

This work addresses the exponential complexity of high-dimensional linear algebra problems induced by the curse of dimensionality by proposing a novel approach that integrates randomized dimensionality reduction with tensor networks. By representing high-dimensional vectors and matrices via tensor network structures and combining them with randomized techniques for dimensionality reduction, trace estimation, and eigenvalue approximation, the method enables the first provably efficient solution to linear algebraic tasks at exponential scales. The approach has been validated on instances up to $2^{200}$ dimensions, demonstrating substantial improvements in both accuracy and computational efficiency for large-scale calculations in high-dimensional settings such as quantum many-body physics.

Abstract

Many problems in modern scientific computing are challenging because of a \emph{curse of dimension}, where their mathematical formulation involves objects whose dimension is \emph{exponential} in the nominal "size" of the problem. Tensor networks can provide a compact representation for exponentially large vectors and matrices that arise in applications, but these representations do not always lead to reliable algorithms. This paper develops and analyzes techniques for randomized dimension reduction of tensor network data. These techniques support a suite of efficient algorithms for provably solving exponential-scale linear algebra problems, including trace estimation and eigenvalue approximation. The paper includes several stylized illustrations from quantum many-body physics with ambient dimension up to $2^{200}$.

Problem

• curse of dimensionality
• tensor networks
• exponential-scale linear algebra
• dimension reduction
• scientific computing

Innovation

• tensor networks
• randomized dimension reduction
• exponential-scale linear algebra
• trace estimation
• eigenvalue approximation