Some of the most successful knowledge graph embedding (KGE) models for link prediction - CP, RESCAL, TUCKER, COMPLEX - can be interpreted as energy-based models. Under this perspective they are not amenable for exact maximum-likelihood estimation (MLE), sampling and struggle to integrate logical constraints. This work re-interprets the score functions of these KGEs as circuits - constrained computational graphs allowing efficient marginalisation. Then, we design two recipes to obtain efficient generative circuit models by either restricting their activations to be non-negative or squaring their outputs. Our interpretation comes with little or no loss of performance for link prediction, while the circuits framework unlocks exact learning by MLE, efficient sampling of new triples, and guarantee that logical constraints are satisfied by design.

A.1 Dataset Details The 20 micro-variations of the 5 macro-variations of the scene were created with the rule of swapping at least two furniture pieces and perturbing the positions of a subset of the other furniture pieces. The occurrences of various furniture objects in these 100 micro-variations are illustrated in Figure 1. Several furniture objects such as'Beanbag' and'Chair' occur more frequently with multiple instances in a some scenes while others such as'Table 03' occur less frequently. We also analyze the object categories of all objects in the original 6 'FRL-apartment' space recreations. We map each of the 92 objects to a semantic category and list the counts per semantic category in a histogram in Figure 1. Since these spaces have a large kitchen area, there is a larger ratio of kitchen objects such as'Kitchen utensil' and'Bowl'. Top down views of the 5 'macro variations' of the scenes are shown in Figure 1. These variations are 5 semantically plausible configurations of furniture in the space generated by a 3D artist. Each surface is annotated with a bounding box, enabling procedural placement of objects on the surfaces. For each of these 5 variations, we generate 20 additional variations, giving 105 scene layouts. Objects are procedurally added on furniture and surfaces using the annotated supporting surface and containment volume information provided by ReplicaCAD.