# Wasserstein-Fisher-Rao Embedding: Logical Query Embeddings with Local Comparison and Global Transport

arXiv: https://arxiv.org/abs/2305.04034 · HTML: https://mlsanity.com/paper/2305.04034/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 39 of its 73 references that are in the map.
- First posted on arXiv 6 May 2023.

## Builds on (most cited first, 30 of 39)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Modeling Relational Data with Graph Convolutional Networks](https://mlsanity.com/paper/1703.06103/) (2017)
- [KGAT: Knowledge Graph Attention Network for Recommendation](https://mlsanity.com/paper/1905.07854/) (2019)
- [DeepPath: A Reinforcement Learning Method for Knowledge Graph Reasoning](https://mlsanity.com/paper/1707.06690/) (2017)
- [Generalized Sliced Wasserstein Distances](https://mlsanity.com/paper/1902.00434/) (2019)
- [Query2box: Reasoning over Knowledge Graphs in Vector Space using Box Embeddings](https://mlsanity.com/paper/2002.05969/) (2020)
- [Beta Embeddings for Multi-Hop Logical Reasoning in Knowledge Graphs](https://mlsanity.com/paper/2010.11465/) (2020)
- [Gromov-Wasserstein Alignment of Word Embedding Spaces](https://mlsanity.com/paper/1809.00013/) (2018)
- [Embedding Logical Queries on Knowledge Graphs](https://mlsanity.com/paper/1806.01445/) (2018)
- [Complex Query Answering with Neural Link Predictors](https://mlsanity.com/paper/2011.03459/) (2020)
- [ConE: Cone Embeddings for Multi-Hop Reasoning over Knowledge Graphs](https://mlsanity.com/paper/2110.13715/) (2021)
- [Sliced Wasserstein Kernel for Persistence Diagrams](https://mlsanity.com/paper/1706.03358/) (2017)
- [Neural-Symbolic Models for Logical Queries on Knowledge Graphs](https://mlsanity.com/paper/2205.10128/) (2022)
- [Vocabulary Learning via Optimal Transport for Neural Machine Translation](https://mlsanity.com/paper/2012.15671/) (2020)
- [Mask and Reason: Pre-Training Knowledge Graph Transformers for Complex Logical Queries](https://mlsanity.com/paper/2208.07638/) (2022)
- [Wasserstein Dictionary Learning: Optimal Transport-based unsupervised non-linear dictionary learning](https://mlsanity.com/paper/1708.01955/) (2017)
- [Self-Supervised Hyperboloid Representations from Logical Queries over Knowledge Graphs](https://mlsanity.com/paper/2012.13023/) (2020)
- [Neural Graph Reasoning: Complex Logical Query Answering Meets Graph Databases](https://mlsanity.com/paper/2303.14617/) (2023)
- [Answering Complex Queries in Knowledge Graphs with Bidirectional Sequence Encoders](https://mlsanity.com/paper/2004.02596/) (2020)
- [Probabilistic Entity Representation Model for Reasoning over Knowledge Graphs](https://mlsanity.com/paper/2110.13522/) (2021)
- [Logical Message Passing Networks with One-hop Inference on Atomic Formulas](https://mlsanity.com/paper/2301.08859/) (2023)
- [Query2Particles: Knowledge Graph Reasoning with Particle Embeddings](https://mlsanity.com/paper/2204.12847/) (2022)
- [Query Embedding on Hyper-relational Knowledge Graphs](https://mlsanity.com/paper/2106.08166/) (2021)
- [Benchmarking the Combinatorial Generalizability of Complex Query Answering on Knowledge Graphs](https://mlsanity.com/paper/2109.08925/) (2021)
- [GammaE: Gamma Embeddings for Logical Queries on Knowledge Graphs](https://mlsanity.com/paper/2210.15578/) (2022)
- [Generalizing Point Embeddings using the Wasserstein Space of Elliptical Distributions](https://mlsanity.com/paper/1805.07594/) (2018)
- [Neural-Symbolic Entangled Framework for Complex Query Answering](https://mlsanity.com/paper/2209.08779/) (2022)
- [Logic Embeddings for Complex Query Answering](https://mlsanity.com/paper/2103.00418/) (2021)
- [SMORE: Knowledge Graph Completion and Multi-hop Reasoning in Massive Knowledge Graphs](https://mlsanity.com/paper/2110.14890/) (2021)
- [Neural Methods for Logical Reasoning Over Knowledge Graphs](https://mlsanity.com/paper/2209.14464/) (2022)

## Built on by (most cited first, 16 of 16)

- [Extending Context Window of Large Language Models via Semantic Compression](https://mlsanity.com/paper/2312.09571/) (2023)
- [Hierarchical Multi-Marginal Optimal Transport for Network Alignment](https://mlsanity.com/paper/2310.04470/) (2023)
- [Top Ten Challenges Towards Agentic Neural Graph Databases](https://mlsanity.com/paper/2501.14224/) (2025)
- [EFOk-CQA: Towards Knowledge Graph Complex Query Answering beyond Set Operation](https://mlsanity.com/paper/2307.13701/) (2023)
- [Efficient Prompt Compression with Evaluator Heads for Long-Context Transformer Inference](https://mlsanity.com/paper/2501.12959/) (2025)
- [Knowledge Graph Reasoning over Entities and Numerical Values](https://mlsanity.com/paper/2306.01399/) (2023)
- [Retrieval Meets Reasoning: Dynamic In-Context Editing for Long-Text Understanding](https://mlsanity.com/paper/2406.12331/) (2024)
- [Privacy-Preserved Neural Graph Databases](https://mlsanity.com/paper/2312.15591/) (2023)
- [Conditional Logical Message Passing Transformer for Complex Query Answering](https://mlsanity.com/paper/2402.12954/) (2024)
- [Query2Triple: Unified Query Encoding for Answering Diverse Complex Queries over Knowledge Graphs](https://mlsanity.com/paper/2310.11246/) (2023)
- [Improving Multi-hop Logical Reasoning in Knowledge Graphs with Context-Aware Query Representation Learning](https://mlsanity.com/paper/2406.07034/) (2024)
- [On the Role of Entity and Event Level Conceptualization in Generalizable Reasoning: A Survey of Tasks, Methods, Applications, and Future Directions](https://mlsanity.com/paper/2406.10885/) (2024)
- [Extending Complex Logical Queries on Uncertain Knowledge Graphs](https://mlsanity.com/paper/2403.01508/) (2024)
- [Meta Operator for Complex Query Answering on Knowledge Graphs](https://mlsanity.com/paper/2403.10110/) (2024)
- [Efficient and Scalable Neural Symbolic Search for Knowledge Graph Complex Query Answering](https://mlsanity.com/paper/2505.08155/) (2025)
- [Neural Scalable Symbolic Search Framework for Complex Logical Queries with Multiple Free Variables](https://mlsanity.com/paper/2605.25985/) (2026)

## Abstract (arXiv)

Answering complex queries on knowledge graphs is important but particularly challenging because of the data incompleteness. Query embedding methods address this issue by learning-based models and simulating logical reasoning with set operators. Previous works focus on specific forms of embeddings, but scoring functions between embeddings are underexplored. In contrast to existing scoring functions motivated by local comparison or global transport, this work investigates the local and global trade-off with unbalanced optimal transport theory. Specifically, we embed sets as bounded measures in real endowed with a scoring function motivated by the Wasserstein-Fisher-Rao metric. Such a design also facilitates closed-form set operators in the embedding space. Moreover, we introduce a convolution-based algorithm for linear time computation and a block-diagonal kernel to enforce the trade-off. Results show that WFRE can outperform existing query embedding methods on standard datasets, evaluation sets with combinatorially complex queries, and hierarchical knowledge graphs. Ablation study shows that finding a better local and global trade-off is essential for performance improvement.