From Tensor Networks to Tractable Circuits, and back

arXiv (Cornell University)

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.

Authors 5

1. Arend-Jan Quist Aachen

   Leiden University

   Affiliation as printed

   Leiden University , Leiden , The Netherlands

2. Marc Farreras Bartra Aachen

   Leiden University · University of Amsterdam

   Affiliation as printed

   Leiden University , Leiden , The Netherlands

   University of Amsterdam , Amsterdam , The Netherlands

3. Alexis de Colnet Aachen

   Leiden University

   Affiliation as printed

   Leiden University , Leiden , The Netherlands

4. John van de Wetering

   University of Amsterdam

   Affiliation as printed

   University of Amsterdam , Amsterdam , The Netherlands

5. Alfons Laarman Aachen

   Leiden University

   Affiliation as printed

   Leiden University , Leiden , The Netherlands