# Choosing the parameter of the Fermat distance: navigating geometry and noise

arXiv: https://arxiv.org/abs/2311.18663 · HTML: https://mlsanity.com/paper/2311.18663/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 29 references that are in the map.
- First posted on arXiv 30 Nov 2023.

## Builds on (most cited first, 7 of 7)

- [Evaluation: from precision, recall and F-measure to ROC, informedness, markedness and correlation](https://mlsanity.com/paper/2010.16061/) (2020)
- [Balancing Geometry and Density: Path Distances on High-Dimensional Data](https://mlsanity.com/paper/2012.09385/) (2020)
- [Fermat Distances: Metric Approximation, Spectral Convergence, and Clustering Algorithms](https://mlsanity.com/paper/2307.05750/) (2023)
- [Power Weighted Shortest Paths for Clustering Euclidean Data](https://mlsanity.com/paper/1905.13345/) (2019)
- [A flexible EM-like clustering algorithm for noisy data](https://mlsanity.com/paper/1907.01660/) (2019)
- [Intrinsic persistent homology via density-based metric learning](https://mlsanity.com/paper/2012.07621/) (2020)
- [Hamilton-Jacobi equations on graphs with applications to semi-supervised learning and data depth](https://mlsanity.com/paper/2202.08789/) (2022)

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

- [Static and Dynamic Approaches to Computing Barycenters of Probability Measures on Graphs](https://mlsanity.com/paper/2603.26940/) (2026)
- [K-means with learned metrics](https://mlsanity.com/paper/2603.14601/) (2026)

## Abstract (arXiv)

The Fermat distance has been recently established as a useful tool for machine learning tasks when a natural distance is not directly available to the practitioner or to improve the results given by Euclidean distances by exploding the geometrical and statistical properties of the dataset. This distance depends on a parameter α that greatly impacts the performance of subsequent tasks. Ideally, the value of α should be large enough to navigate the geometric intricacies inherent to the problem. At the same, it should remain restrained enough to sidestep any deleterious ramifications stemming from noise during the process of distance estimation. We study both theoretically and through simulations how to select this parameter.