Arithmetic circuit tensor networks, multivariable function representation, and high-dimensional integration

TL;DR

This paper introduces a method to convert arithmetic circuits of high-dimensional functions into tensor networks, enabling efficient integration in up to 50 dimensions with better scaling than quasi-Monte Carlo methods.

Contribution

It presents a direct mapping from arithmetic circuits to tensor networks, avoiding optimization, and demonstrates improved high-dimensional integration performance.

Findings

Efficient high-dimensional integration up to 50 dimensions

Favorable cost scaling compared to quasi-Monte Carlo

Example where tensor network structure enables integration impossible with quasi-Monte Carlo

Abstract

Many computational problems can be formulated in terms of high-dimensional functions. Simple representations of such functions and resulting computations with them typically suffer from the "curse of dimensionality", an exponential cost dependence on dimension. Tensor networks provide a way to represent certain classes of high-dimensional functions with polynomial memory. This results in computations where the exponential cost is ameliorated or in some cases, removed, if the tensor network representation can be obtained. Here, we introduce a direct mapping from the arithmetic circuit of a function to arithmetic circuit tensor networks, avoiding the need to perform any optimization or functional fit. We demonstrate the power of the circuit construction in examples of multivariable integration on the unit hypercube in up to 50 dimensions, where the complexity of integration can be understood from the circuit structure. We find very favorable cost scaling compared to quasi-Monte-Carlo integration for these cases, and further give an example where efficient quasi-Monte-Carlo cannot be theoretically performed without knowledge of the underlying tensor network circuit structure.