The Manifold Density Function: An Intrinsic Method for the Validation of Manifold Learning
This work provides a novel validation tool for researchers in manifold learning, addressing a specific bottleneck in unsupervised settings.
The authors tackled the problem of validating manifold learning algorithms by introducing the manifold density function, an intrinsic method that adapts Ripley's K-function to assess how well an algorithm captures latent manifold structure, and they proved convergence and robustness properties.
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.