# Probabilistic Box Embeddings for Uncertain Knowledge Graph Reasoning

arXiv: https://arxiv.org/abs/2104.04597 · HTML: https://mlsanity.com/paper/2104.04597/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 34 references that are in the map.
- First posted on arXiv 9 Apr 2021.

## Builds on (most cited first, 17 of 17)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [ConceptNet 5.5: An Open Multilingual Graph of General Knowledge](https://mlsanity.com/paper/1612.03975/) (2016)
- [Embedding Entities and Relations for Learning and Inference in Knowledge Bases](https://mlsanity.com/paper/1412.6575/) (2014)
- [Complex Embeddings for Simple Link Prediction](https://mlsanity.com/paper/1606.06357/) (2016)
- [RotatE: Knowledge Graph Embedding by Relational Rotation in Complex Space](https://mlsanity.com/paper/1902.10197/) (2019)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [TuckER: Tensor Factorization for Knowledge Graph Completion](https://mlsanity.com/paper/1901.09590/) (2019)
- [Query2box: Reasoning over Knowledge Graphs in Vector Space using Box Embeddings](https://mlsanity.com/paper/2002.05969/) (2020)
- [Hyperbolic Entailment Cones for Learning Hierarchical Embeddings](https://mlsanity.com/paper/1804.01882/) (2018)
- [Order-Embeddings of Images and Language](https://mlsanity.com/paper/1511.06361/) (2015)
- [BoxE: A Box Embedding Model for Knowledge Base Completion](https://mlsanity.com/paper/2007.06267/) (2020)
- [Probabilistic Embedding of Knowledge Graphs with Box Lattice Measures](https://mlsanity.com/paper/1805.06627/) (2018)
- [Hinge-Loss Markov Random Fields and Probabilistic Soft Logic](https://mlsanity.com/paper/1505.04406/) (2015)
- [Joint Constrained Learning for Event-Event Relation Extraction](https://mlsanity.com/paper/2010.06727/) (2020)
- [Improving Local Identifiability in Probabilistic Box Embeddings](https://mlsanity.com/paper/2010.04831/) (2020)
- [Knowledge Association with Hyperbolic Knowledge Graph Embeddings](https://mlsanity.com/paper/2010.02162/) (2020)
- [Embedding Uncertain Knowledge Graphs](https://mlsanity.com/paper/1811.10667/) (2018)

## Built on by (most cited first, 22 of 22)

- [Uncertainty in Natural Language Processing: Sources, Quantification, and Applications](https://mlsanity.com/paper/2306.04459/) (2023)
- [GammaE: Gamma Embeddings for Logical Queries on Knowledge Graphs](https://mlsanity.com/paper/2210.15578/) (2022)
- [How to Turn Your Knowledge Graph Embeddings into Generative Models](https://mlsanity.com/paper/2305.15944/) (2023)
- [Word2Box: Capturing Set-Theoretic Semantics of Words using Box Embeddings](https://mlsanity.com/paper/2106.14361/) (2021)
- [Box Embeddings: An open-source library for representation learning using geometric structures](https://mlsanity.com/paper/2109.04997/) (2021)
- [Geometric Relational Embeddings: A Survey](https://mlsanity.com/paper/2304.11949/) (2023)
- [Certainty in Uncertainty: Reasoning over Uncertain Knowledge Graphs with Statistical Guarantees](https://mlsanity.com/paper/2510.24754/) (2025)
- [Concept2Box: Joint Geometric Embeddings for Learning Two-View Knowledge Graphs](https://mlsanity.com/paper/2307.01933/) (2023)
- [Uncertainty Management in the Construction of Knowledge Graphs: a Survey](https://mlsanity.com/paper/2405.16929/) (2024)
- [Approximating Probabilistic Inference in Statistical EL with Knowledge Graph Embeddings](https://mlsanity.com/paper/2407.11821/) (2024)
- [MolKD: Distilling Cross-Modal Knowledge in Chemical Reactions for Molecular Property Prediction](https://mlsanity.com/paper/2305.01912/) (2023)
- [Noisy Positive-Unlabeled Learning with Self-Training for Speculative Knowledge Graph Reasoning](https://mlsanity.com/paper/2306.07512/) (2023)
- [Extending Complex Logical Queries on Uncertain Knowledge Graphs](https://mlsanity.com/paper/2403.01508/) (2024)
- [Arbitrary Time Information Modeling via Polynomial Approximation for Temporal Knowledge Graph Embedding](https://mlsanity.com/paper/2405.00358/) (2024)
- [FUSE: Measure-Theoretic Compact Fuzzy Set Representation for Taxonomy Expansion](https://mlsanity.com/paper/2506.08409/) (2025)
- [Uncertain Knowledge Graph Completion via Semi-Supervised Confidence Distribution Learning](https://mlsanity.com/paper/2510.16601/) (2025)
- [Set2Box: Similarity Preserving Representation Learning of Sets](https://mlsanity.com/paper/2210.03282/) (2022)
- [Embeddings as Epistemic States: Limitations on the Use of Pooling Operators for Accumulating Knowledge](https://mlsanity.com/paper/2210.05723/) (2022)
- [CAND: Cross-Domain Ambiguity Inference for Early Detecting Nuanced Illness Deterioration](https://mlsanity.com/paper/2501.16365/) (2025)
- [Role Support in Knowledge-Graph Error Ranking: Predictor Regimes and Evaluation Policies](https://mlsanity.com/paper/2512.22318/) (2025)
- [NSFL: A Post-Training Neuro-Symbolic Fuzzy Logic Framework for Boolean Operators in Neural Embeddings](https://mlsanity.com/paper/2604.10604/) (2026)
- [PDPK: A Framework to Synthesise Process Data and Corresponding Procedural Knowledge for Manufacturing](https://mlsanity.com/paper/2308.08371/) (2023)

## Abstract (arXiv)

Knowledge bases often consist of facts which are harvested from a variety of sources, many of which are noisy and some of which conflict, resulting in a level of uncertainty for each triple. Knowledge bases are also often incomplete, prompting the use of embedding methods to generalize from known facts, however, existing embedding methods only model triple-level uncertainty, and reasoning results lack global consistency. To address these shortcomings, we propose BEUrRE, a novel uncertain knowledge graph embedding method with calibrated probabilistic semantics. BEUrRE models each entity as a box (i.e. axis-aligned hyperrectangle) and relations between two entities as affine transforms on the head and tail entity boxes. The geometry of the boxes allows for efficient calculation of intersections and volumes, endowing the model with calibrated probabilistic semantics and facilitating the incorporation of relational constraints. Extensive experiments on two benchmark datasets show that BEUrRE consistently outperforms baselines on confidence prediction and fact ranking due to its probabilistic calibration and ability to capture high-order dependencies among facts.