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Compressing multivariate functions with tree tensor networks

by Joseph Tindall, Miles Stoudenmire and Ryan Levy

Submission summary

Authors (as registered SciPost users):

Joseph Tindall

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.

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• Provide a novel and synergetic link between different research areas.
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• Detail a groundbreaking theoretical/experimental/computational discovery
• Present a breakthrough on a previously-identified and long-standing research stumbling block