Approximating Tensor Network Contraction with Sketches

TL;DR

This paper introduces novel sketching algorithms to efficiently approximate tensor network contractions, including cyclic networks, reducing computational complexity from exponential to polynomial for acyclic cases.

Contribution

It presents the first method capable of approximating arbitrary tensor network contractions and improves efficiency for acyclic networks by reducing complexity.

Findings

First method supports cyclic tensor networks.

Acyclic network approximation has polynomial complexity.

Existing methods require exponential time for general tensor networks.

Abstract

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.