# What relations are reliably embeddable in Euclidean space?

arXiv: https://arxiv.org/abs/1903.05347 · HTML: https://mlsanity.com/paper/1903.05347/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Mar 2019.

## Built on by (most cited first, 6 of 6)

- [Exact Representation of Sparse Networks with Symmetric Nonnegative Embeddings](https://mlsanity.com/paper/2111.03030/) (2021)
- [Binder: Hierarchical Concept Representation through Order Embedding of Binary Vectors](https://mlsanity.com/paper/2404.10924/) (2024)
- [What is Learned in Knowledge Graph Embeddings?](https://mlsanity.com/paper/2110.09978/) (2021)
- [Towards Ordinal Data Science](https://mlsanity.com/paper/2307.09477/) (2023)
- [Relation Geometry in Semantic Space of Language Models](https://mlsanity.com/paper/2607.26762/) (2026)
- [PAGR: Proof-Carrying Algebraic-Geometric Retrieval: A Quiver-, Provenance-, and Sheaf-Theoretic Framework for Grounded LLM Retrieval](https://mlsanity.com/paper/2609.06127/) (2026)

## Abstract (arXiv)

We consider the problem of embedding a relation, represented as a directed graph, into Euclidean space. For three types of embeddings motivated by the recent literature on knowledge graphs, we obtain characterizations of which relations they are able to capture, as well as bounds on the minimal dimensionality and precision needed.