Compressing multivariate functions with tree tensor networks

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.