2024/10/04 by Joseph Tindall, Miles Stoudenmire, Tindall, Joseph +3 · 7 citations
Computer Science · Mathematics · #Computational Physics (physics.comp-ph) #Computational Physics and Python Applications #FOS: Mathematics #FOS: Physical sciences #Numerical Analysis (math.NA) #Quantum Physics (quant-ph) #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.2410.03572
openalex publication_date 2024/10/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.