# Ellipsoid fitting with the Cayley transform

arXiv: https://arxiv.org/abs/2304.10630 · HTML: https://mlsanity.com/paper/2304.10630/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 51 references that are in the map.
- First posted on arXiv 20 Apr 2023.

## Builds on (most cited first, 5 of 5)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Orthogonal Recurrent Neural Networks with Scaled Cayley Transform](https://mlsanity.com/paper/1707.09520/) (2017)
- [Principal Ellipsoid Analysis (PEA): Efficient non-linear dimension reduction & clustering](https://mlsanity.com/paper/2008.07110/) (2020)
- [Concise Complexity Analyses for Trust-Region Methods](https://mlsanity.com/paper/1802.07843/) (2018)
- [Curved factor analysis with the Ellipsoid-Gaussian distribution](https://mlsanity.com/paper/2201.08502/) (2022)

## Abstract (arXiv)

We introduce Cayley transform ellipsoid fitting (CTEF), an algorithm that uses the Cayley transform to fit ellipsoids to noisy data in any dimension. Unlike many ellipsoid fitting methods, CTEF is ellipsoid specific, meaning it always returns elliptic solutions, and can fit arbitrary ellipsoids. It also significantly outperforms other fitting methods when data are not uniformly distributed over the surface of an ellipsoid. Inspired by growing calls for interpretable and reproducible methods in machine learning, we apply CTEF to dimension reduction, data visualization, and clustering in the context of cell cycle and circadian rhythm data and several classical toy examples. Since CTEF captures global curvature, it extracts nonlinear features in data that other machine learning methods fail to identify. For example, on the clustering examples CTEF outperforms 10 popular algorithms.