  We introduce a change of perspective on tensor network states that is defined
by the computational graph of the contraction of an amplitude. The resulting
class of states, which we refer to as tensor network functions, inherit the
conceptual advantages of tensor network states while removing computational
restrictions arising from the need to converge approximate contractions. We use
tensor network functions to compute strict variational estimates of the energy
on loopy graphs, analyze their expressive power for ground-states, show that we
can capture aspects of volume law time evolution, and provide a mapping of
general feed-forward neural nets onto efficient tensor network functions. Our
work expands the realm of computable tensor networks to ones where accurate
contraction methods are not available, and opens up new avenues to use tensor
networks.
