On the Detection of Commutative Factors in Factor Graphs: Necessary and Sufficient Conditions
This work corrects a fundamental error in lifted probabilistic inference, ensuring the reliability of algorithms that exploit object indistinguishability for tractable inference.
The paper identifies a flaw in the state-of-the-art algorithm for detecting commutative factors in factor graphs, where a central theorem was mistakenly treated as sufficient when it is only necessary. The authors provide a corrected theorem and algorithm, ensuring correctness while maintaining efficiency, and introduce a complementary algorithm with tighter worst-case bounds.
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.