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This research demonstrates that tree tensor networks (TTNs) provide a significantly more efficient compressed ansatz for multi-dimensional functions compared to standard tensor trains. By enabling direct constructions of elementary functions and solving non-linear Fredholm equations, the work offers a practical advancement for numerical analysis and computational physics applications. The findings suggest that structured tensor networks could become a preferred tool for high-dimensional data compression and function approximation in scientific computing.
Compressing multivariate functions with tree tensor networks
arXiv:2410.03572v3 Announce Type: replace 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.
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.
Quantum Physics (quant-ph); Numerical Analysis (math.NA); Computational Physics (physics.comp-ph)
arXiv:2410.03572 [quant-ph]
(or arXiv:2410.03572v3 [quant-ph] for this version)
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