On Relaxing Determinism in Arithmetic Circuits

TL;DR

This paper provides a formal framework to compare different variants of arithmetic circuits, clarifies their properties, and presents new theoretical results including separations, completeness, and tractability analyses.

Contribution

It introduces a formal basis for comparing AC variants, clarifies their semantics, and derives new theoretical insights into their properties and limitations.

Findings

Exponential separation between ACs with and without determinism

Completeness and incompleteness results for ACs

Tractability analysis for computing most probable explanations

Abstract

The past decade has seen a significant interest in learning tractable probabilistic representations. Arithmetic circuits (ACs) were among the first proposed tractable representations, with some subsequent representations being instances of ACs with weaker or stronger properties. In this paper, we provide a formal basis under which variants on ACs can be compared, and where the precise roles and semantics of their various properties can be made more transparent. This allows us to place some recent developments on ACs in a clearer perspective and to also derive new results for ACs. This includes an exponential separation between ACs with and without determinism; completeness and incompleteness results; and tractability results (or lack thereof) when computing most probable explanations (MPEs).