# Sailing in high-dimensional spaces: Low-dimensional embeddings through angle preservation

arXiv: https://arxiv.org/abs/2406.09876 · HTML: https://mlsanity.com/paper/2406.09876/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 63 references that are in the map.
- First posted on arXiv 14 Jun 2024.

## Builds on (most cited first, 16 of 16)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Prototypical Contrastive Learning of Unsupervised Representations](https://mlsanity.com/paper/2005.04966/) (2020)
- [Understanding How Dimension Reduction Tools Work: An Empirical Approach to Deciphering t-SNE, UMAP, TriMAP, and PaCMAP for Data Visualization](https://mlsanity.com/paper/2012.04456/) (2020)
- [Supporting Clustering with Contrastive Learning](https://mlsanity.com/paper/2103.12953/) (2021)
- [Efficient Algorithms for t-distributed Stochastic Neighborhood Embedding](https://mlsanity.com/paper/1712.09005/) (2017)
- [Manifold learning: what, how, and why](https://mlsanity.com/paper/2311.03757/) (2023)
- [Theoretical Foundations of t-SNE for Visualizing High-Dimensional Clustered Data](https://mlsanity.com/paper/2105.07536/) (2021)
- [Clustering with t-SNE, provably](https://mlsanity.com/paper/1706.02582/) (2017)
- [Attraction-Repulsion Spectrum in Neighbor Embeddings](https://mlsanity.com/paper/2007.08902/) (2020)
- [On UMAP's true loss function](https://mlsanity.com/paper/2103.14608/) (2021)
- [ScreeNOT: Exact MSE-Optimal Singular Value Thresholding in Correlated Noise](https://mlsanity.com/paper/2009.12297/) (2020)
- [NCVis: Noise Contrastive Approach for Scalable Visualization](https://mlsanity.com/paper/2001.11411/) (2020)
- [A Spectral Method for Assessing and Combining Multiple Data Visualizations](https://mlsanity.com/paper/2210.13711/) (2022)
- [Guarding against Spurious Discoveries in High Dimensions](https://mlsanity.com/paper/1512.01617/) (2015)
- [Are Discoveries Spurious? Distributions of Maximum Spurious Correlations and Their Applications](https://mlsanity.com/paper/1502.04237/) (2015)
- [Preserving local densities in low-dimensional embeddings](https://mlsanity.com/paper/2301.13732/) (2023)

## Built on by (most cited first, 2 of 2)

- [Preserving Angles Improves Feature Distillation](https://mlsanity.com/paper/2411.15239/) (2024)
- [Class Angular Distortion Index for Dimensionality Reduction](https://mlsanity.com/paper/2605.00637/) (2026)

## Abstract (arXiv)

Low-dimensional embeddings (LDEs) of high-dimensional data are ubiquitous in science and engineering. They allow us to quickly understand the main properties of the data, identify outliers and processing errors, and inform the next steps of data analysis. As such, LDEs have to be faithful to the original high-dimensional data, i.e., they should represent the relationships that are encoded in the data, both at a local as well as global scale. The current generation of LDE approaches focus on reconstructing local distances between any pair of samples correctly, often out-performing traditional approaches aiming at all distances. For these approaches, global relationships are, however, usually strongly distorted, often argued to be an inherent trade-off between local and global structure learning for embeddings. We suggest a new perspective on LDE learning, reconstructing angles between data points. We show that this approach, Mercat, yields good reconstruction across a diverse set of experiments and metrics, and preserve structures well across all scales. Compared to existing work, our approach also has a simple formulation, facilitating future theoretical analysis and algorithmic improvements.