# A Metric-based Principal Curve Approach for Learning One-dimensional Manifold

arXiv: https://arxiv.org/abs/2405.12390 · HTML: https://mlsanity.com/paper/2405.12390/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 64 references that are in the map.
- First posted on arXiv 20 May 2024.

## Builds on (most cited first, 9 of 9)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Geoopt: Riemannian Optimization in PyTorch](https://mlsanity.com/paper/2005.02819/) (2020)
- [Pymanopt: A Python Toolbox for Optimization on Manifolds using Automatic Differentiation](https://mlsanity.com/paper/1603.03236/) (2016)
- [geomstats: a Python Package for Riemannian Geometry in Machine Learning](https://mlsanity.com/paper/1805.08308/) (2018)
- [Fr'echet Regression for Random Objects with Euclidean Predictors](https://mlsanity.com/paper/1608.03012/) (2016)
- [Geometrically Enriched Latent Spaces](https://mlsanity.com/paper/2008.00565/) (2020)
- [Wasserstein Regression](https://mlsanity.com/paper/2006.09660/) (2020)
- [A Locally Adaptive Normal Distribution](https://mlsanity.com/paper/1606.02518/) (2016)
- [Invertible Kernel PCA with Random Fourier Features](https://mlsanity.com/paper/2303.05043/) (2023)

## Abstract (arXiv)

Principal curve is a well-known statistical method oriented in manifold learning using concepts from differential geometry. In this paper, we propose a novel metric-based principal curve (MPC) method that learns one-dimensional manifold of spatial data. Synthetic datasets Real applications using MNIST dataset show that our method can learn the one-dimensional manifold well in terms of the shape.