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Peer-reviewedCC BY 4.0SciPost Physics

Compressing multivariate functions with tree tensor networks

  • Joseph Tindall1
  • Miles Stoudenmire1
  • Ryan Levy1,2
  1. 1Flatiron Institute
  2. 2PsiQuantum (United States)
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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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Cite

BibTeXRIS
BibTeX
@article{tindall2026compressing,
  title = {{Compressing multivariate functions with tree tensor networks}},
  author = {Joseph Tindall and Miles Stoudenmire and Ryan Levy},
  journal = {SciPost Physics},
  year = {2026},
  month = oct,
  doi = {10.21468/scipostphys.21.4.095},
  url = {https://scipost.org/10.21468/SciPostPhys.21.4.095},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Compressing multivariate functions with tree tensor networks
AU  - Joseph Tindall
AU  - Miles Stoudenmire
AU  - Ryan Levy
PY  - 2026
DA  - 2026/10/15
JO  - SciPost Physics
DO  - 10.21468/scipostphys.21.4.095
UR  - https://scipost.org/10.21468/SciPostPhys.21.4.095
AB  - 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.
SN  - 2542-4653
ER  -