# Improving Local Identifiability in Probabilistic Box Embeddings

arXiv: https://arxiv.org/abs/2010.04831 · HTML: https://mlsanity.com/paper/2010.04831/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Oct 2020.

## Built on by (most cited first, 30 of 35)

- [Hyperbolic Image-Text Representations](https://mlsanity.com/paper/2304.09172/) (2023)
- [ConE: Cone Embeddings for Multi-Hop Reasoning over Knowledge Graphs](https://mlsanity.com/paper/2110.13715/) (2021)
- [Modeling Heterogeneous Hierarchies with Relation-specific Hyperbolic Cones](https://mlsanity.com/paper/2110.14923/) (2021)
- [Modeling Fine-Grained Entity Types with Box Embeddings](https://mlsanity.com/paper/2101.00345/) (2021)
- [ExACT: Language-guided Conceptual Reasoning and Uncertainty Estimation for Event-based Action Recognition and More](https://mlsanity.com/paper/2403.12534/) (2024)
- [Faithiful Embeddings for EL++ Knowledge Bases](https://mlsanity.com/paper/2201.09919/) (2022)
- [Probabilistic Box Embeddings for Uncertain Knowledge Graph Reasoning](https://mlsanity.com/paper/2104.04597/) (2021)
- [Learning with Algorithmic Supervision via Continuous Relaxations](https://mlsanity.com/paper/2110.05651/) (2021)
- [GammaE: Gamma Embeddings for Logical Queries on Knowledge Graphs](https://mlsanity.com/paper/2210.15578/) (2022)
- [Ontology Embedding: A Survey of Methods, Applications and Resources](https://mlsanity.com/paper/2406.10964/) (2024)
- [Interpretable Text Embeddings and Text Similarity Explanation: A Survey](https://mlsanity.com/paper/2502.14862/) (2025)
- [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)
- [PHyCLIP: ℓ1-Product of Hyperbolic Factors Unifies Hierarchy and Compositionality in Vision-Language Representation Learning](https://mlsanity.com/paper/2510.08919/) (2025)
- [Concept2Box: Joint Geometric Embeddings for Learning Two-View Knowledge Graphs](https://mlsanity.com/paper/2307.01933/) (2023)
- [Representing Spatial Trajectories as Distributions](https://mlsanity.com/paper/2210.01322/) (2022)
- [Approximating Probabilistic Inference in Statistical EL with Knowledge Graph Embeddings](https://mlsanity.com/paper/2407.11821/) (2024)
- [Self-supervised Topic Taxonomy Discovery in the Box Embedding Space](https://mlsanity.com/paper/2408.15050/) (2024)
- [Binder: Hierarchical Concept Representation through Order Embedding of Binary Vectors](https://mlsanity.com/paper/2404.10924/) (2024)
- [Polar Ducks and Where to Find Them: Enhancing Entity Linking with Duck Typing and Polar Box Embeddings](https://mlsanity.com/paper/2305.12027/) (2023)
- [Arbitrary Time Information Modeling via Polynomial Approximation for Temporal Knowledge Graph Embedding](https://mlsanity.com/paper/2405.00358/) (2024)
- [TaxoBell: Gaussian Box Embeddings for Self-Supervised Taxonomy Expansion](https://mlsanity.com/paper/2601.09633/) (2026)
- [Set2Box: Similarity Preserving Representation Learning of Sets](https://mlsanity.com/paper/2210.03282/) (2022)
- [Learning Low-Rank Latent Spaces with Simple Deterministic Autoencoder: Theoretical and Empirical Insights](https://mlsanity.com/paper/2310.16194/) (2023)
- [When Box Meets Graph Neural Network in Tag-aware Recommendation](https://mlsanity.com/paper/2406.12020/) (2024)
- [RegD: Hierarchical Embeddings via Dissimilarity between Arbitrary Euclidean Regions](https://mlsanity.com/paper/2501.17518/) (2025)
- [Graph2Region: Efficient Graph Similarity Learning with Structure and Scale Restoration](https://mlsanity.com/paper/2510.00394/) (2025)
- [Polaris: Coupled Orbital Polar Embeddings for Hierarchical Concept Learning](https://mlsanity.com/paper/2605.00265/) (2026)
- [Graph Neural Network based Hierarchy-Aware Embeddings of Knowledge Graphs: Applications to Yeast Phenotype Prediction](https://mlsanity.com/paper/2605.03690/) (2026)

## Abstract (arXiv)

Geometric embeddings have recently received attention for their natural ability to represent transitive asymmetric relations via containment. Box embeddings, where objects are represented by n-dimensional hyperrectangles, are a particularly promising example of such an embedding as they are closed under intersection and their volume can be calculated easily, allowing them to naturally represent calibrated probability distributions. The benefits of geometric embeddings also introduce a problem of local identifiability, however, where whole neighborhoods of parameters result in equivalent loss which impedes learning. Prior work addressed some of these issues by using an approximation to Gaussian convolution over the box parameters, however, this intersection operation also increases the sparsity of the gradient. In this work, we model the box parameters with min and max Gumbel distributions, which were chosen such that space is still closed under the operation of the intersection. The calculation of the expected intersection volume involves all parameters, and we demonstrate experimentally that this drastically improves the ability of such models to learn.