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

2024/02/14 by Holmgren, Benjamin, Quist, Eli, Schupbach, Jordan +2
#57Z25 #Algebraic Topology (math.AT) #FOS: Computer and information sciences #FOS: Mathematics #I.5.2 #Machine Learning (cs.LG)

paper · doi:10.48550/arxiv.2402.09529

Abstract

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.

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