# Unsupervised Embedding of Hierarchical Structure in Euclidean Space

arXiv: https://arxiv.org/abs/2010.16055 · HTML: https://mlsanity.com/paper/2010.16055/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Oct 2020.

## Built on by (most cited first, 3 of 3)

- [Unsupervised Opinion Summarization Using Approximate Geodesics](https://mlsanity.com/paper/2209.07496/) (2022)
- [Enhancing Group Fairness in Online Settings Using Oblique Decision Forests](https://mlsanity.com/paper/2310.11401/) (2023)
- [Hyperbolic Gaussian Blurring Mean Shift: A Statistical Mode-Seeking Framework for Clustering in Curved Spaces](https://mlsanity.com/paper/2512.11448/) (2025)

## Abstract (arXiv)

Deep embedding methods have influenced many areas of unsupervised learning. However, the best methods for learning hierarchical structure use non-Euclidean representations, whereas Euclidean geometry underlies the theory behind many hierarchical clustering algorithms. To bridge the gap between these two areas, we consider learning a non-linear embedding of data into Euclidean space as a way to improve the hierarchical clustering produced by agglomerative algorithms. To learn the embedding, we revisit using a variational autoencoder with a Gaussian mixture prior, and we show that rescaling the latent space embedding and then applying Ward's linkage-based algorithm leads to improved results for both dendrogram purity and the Moseley-Wang cost function. Finally, we complement our empirical results with a theoretical explanation of the success of this approach. We study a synthetic model of the embedded vectors and prove that Ward's method exactly recovers the planted hierarchical clustering with high probability.