Approximating Tensor Network Contraction with Sketches
This work provides more efficient approximation methods for tensor network contractions, which is a critical problem for researchers and practitioners in fields like machine learning, quantum mechanics, and database systems. It offers a significant improvement for a broad range of applications.
The paper addresses the computational expense of tensor network contractions, which are fundamental operations in various fields. It introduces the first method to approximate arbitrary tensor network contractions, including cyclic ones. Additionally, it presents a new method for acyclic tensor networks that achieves polynomial space and time complexity, improving upon existing methods that exhibit exponential complexity with the number of contractions.
Tensor network contraction is a fundamental mathematical operation that generalizes the dot product and matrix multiplication. It finds applications in numerous domains, such as database systems, graph theory, machine learning, probability theory, and quantum mechanics. Tensor network contractions are computationally expensive, in general requiring exponential time and space. Sketching methods include a number of dimensionality reduction techniques that are widely used in the design of approximation algorithms. The existing sketching methods for tensor network contraction, however, only support acyclic tensor networks. We present the first method capable of approximating arbitrary tensor network contractions, including those of cyclic tensor networks. Additionally, we show that the existing sketching methods require a computational complexity that grows exponentially with the number of contractions. We present a second method, for acyclic tensor networks, whose space and time complexity depends only polynomially on the number of contractions.