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arXiv · 2410.03572

Compressing multivariate functions with tree tensor networks

Joseph Tindall·E. Miles Stoudenmire·Ryan Levy
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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 (TTNs) 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. Finally, we demonstrate how the methods introduced in this work can be used to realize a TTN-based solver for multi-dimensional, non-linear Fredholm equations.

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Joseph Tindall, E. Miles Stoudenmire, Ryan Levy. 2024-10-04. Compressing multivariate functions with tree tensor networks. https://arxiv.org/abs/2410.03572

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