Interaction decomposition for Hilbert spaces

TL;DR

This paper introduces a unified framework for interaction decomposition in Hilbert spaces, extending concepts from graphical models and statistical physics to a categorical setting.

Contribution

It defines and characterizes decomposable functors from posets to Hilbert spaces, unifying and extending existing interaction decompositions.

Findings

Unified framework for interaction decomposition in Hilbert spaces

Extension of previous work on decomposable collections and presheaves

Characterization of decomposable functors from posets to Hilbert spaces

Abstract

The decomposition into interaction subspaces is an important result for graphical models and plays a central role for results on the linearized marginal problem; similarly the Chaos decomposition plays an important role in statistical physics at the thermodynamic limit and in probability theory in general. We unify and extend both constructions by defining and characterizing decomposable functors from a well-founded poset to the category of Hilbert spaces, completing previous work on decomposable collections of vector subspaces and presheaves.