From Tensor Networks to Tractable Circuits, and back

TL;DR

This paper establishes formal equivalences between tensor networks and circuits, enabling the transfer of structural and algorithmic results across these widely used data structures.

Contribution

It proves that certain classes of tensor networks correspond exactly to classes of tractable circuits, bridging two research communities.

Findings

Matrix product states coincide with nondeterministic edge-valued decision diagrams.

Tree tensor networks correspond to structured-decomposable circuits.

Structural and algorithmic guarantees can be transferred between tensor networks and circuits.

Abstract

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.