Compressing multivariate functions with tree tensor networks

TL;DR

This paper explores the use of tree tensor networks to efficiently compress and approximate multivariate functions, outperforming tensor trains in certain applications and enabling solutions to complex multi-dimensional equations.

Contribution

It introduces the use of general tree tensor networks for function approximation, providing new constructions and algorithms that improve efficiency over tensor trains.

Findings

Tree tensor networks can represent elementary functions efficiently.

Structured tree tensor networks outperform tensor trains for multi-dimensional functions.

Application to non-linear Fredholm equations shows exponential accuracy scaling.

Abstract

Tensor networks are a compressed format for multi-dimensional data. One-dimensional tensor networks -- often referred to as tensor trains (TT) or matrix product states (MPS) -- are increasingly being used as a numerical ansatz for continuum functions by ``quantizing'' the inputs into discrete binary digits. Here we demonstrate the power of more general tree tensor networks for this purpose. We provide direct constructions of a number of elementary functions as generic tree tensor networks and interpolative constructions for more complicated functions via a generalization of the tensor cross interpolation algorithm. For a range of multi-dimensional functions we show how more structured tree tensor networks offer a significantly more efficient ansatz than the commonly used tensor train. We demonstrate an application of our methods to solving multi-dimensional, non-linear Fredholm equations, providing a rigorous bound on the rank of the solution which, in turn, guarantees exponentially scaling accuracy with the size of the tree tensor network for certain problems.