# Hierarchical Density Order Embeddings

arXiv: https://arxiv.org/abs/1804.09843 · HTML: https://mlsanity.com/paper/1804.09843/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Apr 2018.

## Built on by (most cited first, 13 of 13)

- [CONTaiNER: Few-Shot Named Entity Recognition via Contrastive Learning](https://mlsanity.com/paper/2109.07589/) (2021)
- [Emergent Visual-Semantic Hierarchies in Image-Text Representations](https://mlsanity.com/paper/2407.08521/) (2024)
- [Ontology Embedding: A Survey of Methods, Applications and Resources](https://mlsanity.com/paper/2406.10964/) (2024)
- [Modeling Layout Reading Order as Ordering Relations for Visually-rich Document Understanding](https://mlsanity.com/paper/2409.19672/) (2024)
- [Word2Box: Capturing Set-Theoretic Semantics of Words using Box Embeddings](https://mlsanity.com/paper/2106.14361/) (2021)
- [Improving Visual Recognition with Hyperbolical Visual Hierarchy Mapping](https://mlsanity.com/paper/2404.00974/) (2024)
- [Box Embeddings: An open-source library for representation learning using geometric structures](https://mlsanity.com/paper/2109.04997/) (2021)
- [TopicNet: Semantic Graph-Guided Topic Discovery](https://mlsanity.com/paper/2110.14286/) (2021)
- [Hierarchical Retrieval: The Geometry and a Pretrain-Finetune Recipe](https://mlsanity.com/paper/2509.16411/) (2025)
- [Uncertainty-based Cross-Modal Retrieval with Probabilistic Representations](https://mlsanity.com/paper/2204.09268/) (2022)
- [HYDEN: Hyperbolic Density Representations for Medical Images and Reports](https://mlsanity.com/paper/2408.09715/) (2024)
- [Native Hierarchical and Compositional Representations with Subspace Embeddings](https://mlsanity.com/paper/2508.16687/) (2025)
- [A Differentiable Bayesian Relaxation for Latent Partial-Order Inference](https://mlsanity.com/paper/2605.06976/) (2026)

## Abstract (arXiv)

By representing words with probability densities rather than point vectors, probabilistic word embeddings can capture rich and interpretable semantic information and uncertainty. The uncertainty information can be particularly meaningful in capturing entailment relationships -- whereby general words such as "entity" correspond to broad distributions that encompass more specific words such as "animal" or "instrument". We introduce density order embeddings, which learn hierarchical representations through encapsulation of probability densities. In particular, we propose simple yet effective loss functions and distance metrics, as well as graph-based schemes to select negative samples to better learn hierarchical density representations. Our approach provides state-of-the-art performance on the WordNet hypernym relationship prediction task and the challenging HyperLex lexical entailment dataset -- while retaining a rich and interpretable density representation.