DECOR: Detection of Commutative Factors

• DECOR is an algorithm for detecting commutative factors by identifying exchangeable argument subsets in factor graphs to leverage internal symmetry in lifted inference.
• It replaces the exponential O(2^n) search with a bucket-based restriction mechanism that derives candidate subsets from duplicate patterns, reducing computational overhead.
• The method improves lifted model construction by enabling efficient message passing and grouping of indistinguishable variables, thereby streamlining probabilistic inference.

In probabilistic graphical models, DECOR denotes the Detection of Commutative Factors algorithm, introduced to identify exchangeable argument sets inside factors of a factor graph so that lifted probabilistic inference can exploit internal symmetries during lifted model construction (Luttermann et al., 2024). The method addresses a specific bottleneck: prior commutativity detection in the ACP, or CPR, pipeline checked candidate argument subsets by descending-size enumeration, which can require $O(2^n)$ subset iterations for a factor with $n$ arguments in the worst case. DECOR replaces that search with a bucket-based restriction mechanism that derives candidate commutative subsets from duplicate potential patterns and prunes the search dramatically in practice (Luttermann et al., 2024).

1. Factor graphs, lifted inference, and the role of commutative factors

The setting is a factor graph defined as an undirected bipartite graph

$G = (\boldsymbol V, \boldsymbol E)$

with

$$\boldsymbol V = \boldsymbol R \cup \boldsymbol \Phi,$$

where $\boldsymbol R = \{R_1, \ldots, R_n\}$ are variable nodes and $\boldsymbol \Phi = \{\phi_1, \ldots, \phi_m\}$ are factor nodes. Its semantics is

$$P_G = \frac{1}{Z} \prod_{j=1}^m \phi_j(\mathcal A_j),$$

where $Z$ is the normalization constant and $\mathcal A_j$ is the tuple of random variables adjacent to factor $\phi_j$ (Luttermann et al., 2024).

The motivation for DECOR comes from lifted probabilistic inference. Lifted methods exploit symmetry so they can reason over groups of indistinguishable individuals rather than each ground variable separately. In the factor-graph setting considered here, one important symmetry is internal to a single factor: some argument positions may be interchangeable without changing the factor value. These are the paper’s commutative factors. Detecting them matters because the advanced colour passing, or ACP/CPR, algorithm for lifted model construction needs to know whether a factor can send the same message to several neighboring variables irrespective of their argument positions (Luttermann et al., 2024).

If arguments are exchangeable, they can be grouped in the lifted representation, often via counting random variables. This reduces representation size and inference cost. A plausible implication is that commutative-factor detection is not merely a preprocessing convenience; it determines whether internal factor symmetry is available to the lifted representation at all.

2. Formal definition of commutativity and the baseline problem

For a factor $n$0, the factor is commutative with respect to a subset $n$1 iff for all events $n$2,

$n$3

for all permutations $n$4 of $n$5 such that \