On the Detection of Commutative Factors in Factor Graphs: Necessary and Sufficient Conditions

2026-05-26

AI Summary

This work addresses a critical flaw in existing exchangeable factor detection algorithms, which rely on a theorem that provides only a necessary condition—erroneously treated as sufficient—potentially leading to incorrect inferences. The paper rigorously clarifies the necessary and sufficient conditions for exchangeability, thereby rectifying this theoretical deficiency. Building on this corrected foundation, two novel algorithms are proposed: one guarantees both correctness and computational efficiency, while the other achieves a tighter worst-case complexity bound. By integrating probabilistic graphical models, permutation invariance theory, and formal verification, this study establishes a theoretically sound and practically scalable approach to detecting exchangeable factors.

Abstract

Exploiting the indistinguishability of objects in a probabilistic graphical model such as a factor graph is key to lifted probabilistic inference algorithms and allows for tractable probabilistic inference problems with respect to domain sizes. A central building block for the exploitation of indistinguishable objects in factor graphs is the identification of commutative factors, i.e., factors whose output values are invariant under permutations of input values assigned to a subset of their arguments. In this paper, we revisit the theoretical foundations underlying the state-of-the-art algorithm to detect commutative factors. Specifically, we show that in its current form, the state-of-the-art algorithm relies on a central theorem that is mistakenly regarded as a sufficient condition to identify commutative factors, while it actually only implies necessary condition. Consequently, the state of the art might, as we show in this paper, deliver incorrect results. To fix the flaws currently present in the state of the art, we prove a slightly modified version of the aforementioned theorem, which serves as a necessary condition to identify commutative factors. Moreover, we present a corrected version of the state-of-the-art algorithm, which keeps its efficiency while ensuring correctness and introduce a complementary algorithm with tighter worst-case bounds.

Problem

• commutative factors
• factor graphs
• lifted probabilistic inference
• necessary and sufficient conditions
• probabilistic graphical models

Innovation

• commutative factors
• factor graphs
• lifted inference
• necessary and sufficient conditions
• probabilistic graphical models