# The Manifold Density Function: An Intrinsic Method for the Validation of Manifold Learning

arXiv: https://arxiv.org/abs/2402.09529 · HTML: https://mlsanity.com/paper/2402.09529/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 38 references that are in the map.
- First posted on arXiv 14 Feb 2024.

## Builds on (most cited first, 3 of 3)

- [Functional summary statistics for point processes on the sphere with an application to determinantal point processes](https://mlsanity.com/paper/1601.03448/) (2016)
- [Geometric Learning and Topological Inference with Biobotic Networks: Convergence Analysis](https://mlsanity.com/paper/1607.00051/) (2016)
- [Generalizations of Ripley's K-function with Application to Space Curves](https://mlsanity.com/paper/1812.06870/) (2018)

## Abstract (arXiv)

We introduce the manifold density function, which is an intrinsic method to validate manifold learning techniques. Our approach adapts and extends Ripley's K-function, and categorizes in an unsupervised setting the extent to which an output of a manifold learning algorithm captures the structure of a latent manifold. Our manifold density function generalizes to broad classes of Riemannian manifolds. In particular, we extend the manifold density function to general two-manifolds using the Gauss-Bonnet theorem, and demonstrate that the manifold density function for hypersurfaces is well approximated using the first Laplacian eigenvalue. We prove desirable convergence and robustness properties.