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Lifted Model Construction without Normalisation: A Vectorised Approach to Exploit Symmetries in Factor Graphs

Malte Luttermann, Ralf Möller, Marcel Gehrke

TL;DR

This paper proposes a generalisation of the advanced colour passing (ACP) algorithm, which is the state of the art to construct a parametric factor graph, that allows for potentials of factors to be scaled arbitrarily and efficiently detects more symmetries than the original ACP.

Abstract

Lifted probabilistic inference exploits symmetries in a probabilistic model to allow for tractable probabilistic inference with respect to domain sizes of logical variables. We found that the current state-of-the-art algorithm to construct a lifted representation in form of a parametric factor graph misses symmetries between factors that are exchangeable but scaled differently, thereby leading to a less compact representation. In this paper, we propose a generalisation of the advanced colour passing (ACP) algorithm, which is the state of the art to construct a parametric factor graph. Our proposed algorithm allows for potentials of factors to be scaled arbitrarily and efficiently detects more symmetries than the original ACP algorithm. By detecting strictly more symmetries than ACP, our algorithm significantly reduces online query times for probabilistic inference when the resulting model is applied, which we also confirm in our experiments.

Lifted Model Construction without Normalisation: A Vectorised Approach to Exploit Symmetries in Factor Graphs

TL;DR

This paper proposes a generalisation of the advanced colour passing (ACP) algorithm, which is the state of the art to construct a parametric factor graph, that allows for potentials of factors to be scaled arbitrarily and efficiently detects more symmetries than the original ACP.

Abstract

Lifted probabilistic inference exploits symmetries in a probabilistic model to allow for tractable probabilistic inference with respect to domain sizes of logical variables. We found that the current state-of-the-art algorithm to construct a lifted representation in form of a parametric factor graph misses symmetries between factors that are exchangeable but scaled differently, thereby leading to a less compact representation. In this paper, we propose a generalisation of the advanced colour passing (ACP) algorithm, which is the state of the art to construct a parametric factor graph. Our proposed algorithm allows for potentials of factors to be scaled arbitrarily and efficiently detects more symmetries than the original ACP algorithm. By detecting strictly more symmetries than ACP, our algorithm significantly reduces online query times for probabilistic inference when the resulting model is applied, which we also confirm in our experiments.

Paper Structure

This paper contains 12 sections, 4 theorems, 11 equations, 10 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Avoiding Normalisation During Lifted Model Construction
  4. Dealing with Scaled Potentials
  5. Dealing with Permutations of Arguments
  6. Experiments
  7. Conclusion
  8. Probabilistic Inference in More Detail
  9. Formal Description of the Advanced Colour Passing Algorithm
  10. Missing Proofs
  11. Checking Collinearity of Vectors without Division Operations
  12. Additional Experimental Results

Key Result

Theorem 1

Let $G = (\boldsymbol V, \boldsymbol E)$ denote an fg with $\boldsymbol V = \boldsymbol R \cup \boldsymbol \Phi$, where $\boldsymbol R = \{R_1, \ldots, R_n\}$ is a set of rv and $\boldsymbol \Phi = \{\phi_1, \ldots, \phi_m\}$ is a set of factors. Then, scaling any factor $\phi_k \in \boldsymbol \Phi

Figures (10)

  • Figure 1: (a) An fg encoding a full joint probability distribution for an epidemic example Hoffmann2022a, (b) a pfg corresponding to the lifted representation of the fg shown in \ref{['fig:example_fg_epid']}. The mappings of argument values to potentials of the factors are omitted for brevity.
  • Figure 2: (a) An exemplary fg, (b) another fg encoding equivalent semantics as the fg shown in (a) but containing a factor $\phi_2$ whose potentials are scaled by factor $\alpha \in \mathbb{R}^+$.
  • Figure 3: Vector representations for exemplary factors $\phi_1, \ldots, \phi_4$. For the sake of this example, every factor maps two possible assignments to a potential value each. The mappings of the factors are encoded as vectors and are given by $\vec{\phi}_1 = (8, 2)$ (i.e., $\phi_1$ maps its first assignment to potential value $8$ and the second assignment to potential value $2$), $\vec{\phi}_2 = (4, 1)$, $\vec{\phi}_3 = (2, 2)$, and $\vec{\phi}_4 = (4.4, 3.6)$.
  • Figure 4: An fg entailing equivalent semantics as the fg shown in \ref{['fig:example_fg', 'fig:example_fg_scaled']} with corresponding buckets. Note that the arguments of the factor $\phi_2$ are arranged in a different order than in \ref{['fig:example_fg', 'fig:example_fg_scaled']} as $B$ appears at position one and $C$ at position two, whereas in the previous examples, $C$ was at position one and $B$ at position two.
  • Figure 5: Average query times of lve on the output of acp and aacp (left) and the average number $\beta$ of queries after which the offline overhead of aacp amortises (right).
  • ...and 5 more figures

Theorems & Definitions (23)

  • Definition 1: Factor Graph
  • Example 1
  • Example 2
  • Example 3
  • Definition 2: Exchangeable Factors
  • Example 4
  • Theorem 1
  • proof
  • Definition 3: Vector Representation of Factors
  • Example 5
  • ...and 13 more