The Manifold Density Function: An Intrinsic Method for the Validation of Manifold Learning
<|reference_start|>The Manifold Density Function: An Intrinsic Method for the Validation of Manifold Learning: 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.<|reference_end|>
@article{holmgren2024the,
    title={The Manifold Density Function: An Intrinsic Method for the Validation of
  Manifold Learning},
    author={Benjamin Holmgren, Eli Quist, Jordan Schupbach, Brittany Terese Fasy,
  Bastian Rieck},
    journal={arXiv preprint arXiv:2402.09529},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.09529},
    primaryClass={cs.LG math.AT}
}