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Fully Geometric Multi-Hop Reasoning on Knowledge Graphs with Transitive Relations

Fernando Zhapa-Camacho, Robert Hoehndorf

TL;DR

GeometrE presents a purely geometric, box-embedding approach for multi-hop reasoning on knowledge graphs, enabling interpretable query operations by mapping logical operators to fixed geometric transformations. A transitive loss is introduced to preserve transitivity in the embedding space, supporting chain-like reasoning across multiple hops. Empirical results on WN18RR-QA, NELL-QA, and FB237 show competitive or superior performance to state-of-the-art baselines, with ablations demonstrating the value of combined additive/multiplicative projections and answer-embedding choices. The work advances interpretable KG reasoning and offers a practical framework for integrating complex query types while preserving geometric interpretations, though negation approximation remains a limitation and a direction for future work.

Abstract

Geometric embedding methods have shown to be useful for multi-hop reasoning on knowledge graphs by mapping entities and logical operations to geometric regions and geometric transformations, respectively. Geometric embeddings provide direct interpretability framework for queries. However, current methods have only leveraged the geometric construction of entities, failing to map logical operations to geometric transformations and, instead, using neural components to learn these operations. We introduce GeometrE, a geometric embedding method for multi-hop reasoning, which does not require learning the logical operations and enables full geometric interpretability. Additionally, unlike previous methods, we introduce a transitive loss function and show that it can preserve the logical rule $\forall a,b,c: r(a,b) \land r(b,c) \to r(a,c)$. Our experiments show that GeometrE outperforms current state-of-the-art methods on standard benchmark datasets.

Fully Geometric Multi-Hop Reasoning on Knowledge Graphs with Transitive Relations

TL;DR

GeometrE presents a purely geometric, box-embedding approach for multi-hop reasoning on knowledge graphs, enabling interpretable query operations by mapping logical operators to fixed geometric transformations. A transitive loss is introduced to preserve transitivity in the embedding space, supporting chain-like reasoning across multiple hops. Empirical results on WN18RR-QA, NELL-QA, and FB237 show competitive or superior performance to state-of-the-art baselines, with ablations demonstrating the value of combined additive/multiplicative projections and answer-embedding choices. The work advances interpretable KG reasoning and offers a practical framework for integrating complex query types while preserving geometric interpretations, though negation approximation remains a limitation and a direction for future work.

Abstract

Geometric embedding methods have shown to be useful for multi-hop reasoning on knowledge graphs by mapping entities and logical operations to geometric regions and geometric transformations, respectively. Geometric embeddings provide direct interpretability framework for queries. However, current methods have only leveraged the geometric construction of entities, failing to map logical operations to geometric transformations and, instead, using neural components to learn these operations. We introduce GeometrE, a geometric embedding method for multi-hop reasoning, which does not require learning the logical operations and enables full geometric interpretability. Additionally, unlike previous methods, we introduce a transitive loss function and show that it can preserve the logical rule . Our experiments show that GeometrE outperforms current state-of-the-art methods on standard benchmark datasets.
Paper Structure (35 sections, 5 theorems, 30 equations, 3 figures, 11 tables)

This paper contains 35 sections, 5 theorems, 30 equations, 3 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Geometric interpretability.
  3. Modeling of transitive relations.
  4. Related Work
  5. Geometric embeddings:
  6. Embeddings for transitive relations.
  7. Preliminaries
  8. Knowledge Graphs.
  9. Logical queries.
  10. Relational properties
  11. Methodology
  12. Box.
  13. Geometric projection.
  14. Geometric Intersection.
  15. Geometric negation.
  16. ...and 20 more sections

Key Result

Theorem 1

Let $(a,r,b), (b,r,c) \in \mathcal{E}$ with $r$ being a transitive relation, but $(a,r,c) \not\in \mathcal{E}$. Let query and answer embeddings for $a,b,c$ and relation embedding for $r$ be optimized using the loss function $L' = L + L_{\hat{{\bm{r}}}_{i}}$ until convergence. Let $i$ be the dimensio

Figures (3)

  • Figure 1: GeometrE representation of boxes and logical operations in $\mathbb{R}^2$.
  • Figure 2: Query types supported by GeometrE. Blue nodes represent anchor entities. Gray nodes represent intermediate operations (relation projection, intersection, union, negation) and green nodes represent answer entities.
  • Figure 3: Transitive chains. For a chain $a_1,\ldots,a_n$ we plot a green line if the embeddings $emb(a_i)>emb(a_{i+1})$, otherwise we plot a red line. Upper plots show the embeddings generated using the transitive loss and the lower plots show the embeddings generated using the normal loss function.

Theorems & Definitions (10)

  • Theorem 1: Transitive Inference Property
  • proof
  • Lemma 1: Idempotency of box transformation
  • proof
  • Theorem 2: Box Complement Intersection
  • proof
  • Theorem 3: Probability of Non-Disjoint Boxes
  • proof
  • Theorem 4: Dimensional effect on box overlap probability
  • proof