From Tensor Networks to Tractable Circuits, and back
For researchers in tensor networks and knowledge compilation, this bridges two separate communities by proving equivalences that allow direct transfer of canonicity and tractability guarantees.
This paper establishes formal equivalences between tensor networks (matrix product states and tree tensor networks) and tractable circuit classes (nondeterministic edge-valued decision diagrams and structured-decomposable circuits), enabling cross-transfer of structural and algorithmic results.
Tensor networks and circuits are widely used data structures to represent pseudo-Boolean functions. These two formalisms have been studied primarily in separate communities, and this paper aims to establish equivalences between them. We show that some classes of tensor networks that are appealing in practice correspond to classes of circuits with specific properties that have been studied in knowledge compilation as \emph{tractable circuits}. In particular, we prove that matrix product states (tensor trains) coincide with nondeterministic edge-valued decision diagrams and that tree tensor networks exactly correspond to structured-decomposable circuits. These correspondences enable direct transfer of structural and algorithmic results; for example, canonicity and tractability guarantees known for circuits yield analogous guarantees for the associated tensor networks, and vice versa.