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

TL;DR

The paper tackles intrinsic validation of manifold learning in the absence of ground-truth geodesic distances by introducing the manifold density function $K_{ abla X}$, extending Ripley's $K$-function to general Riemannian manifolds. It develops local and aggregated estimators $\ ilde{K}$ and defines manifold scores $M_p$ and $M$ to quantify how well a learned representation preserves latent manifold structure, with convergence guarantees for small radii and large samples. For curved two-manifolds, aggregation leverages the Gauss-Bonnet theorem and Euler characteristic $ ext{χ}( abla X)$; in higher dimensions, aggregation is scaled by the first Laplacian eigenvalue $oldsymbol{ abla}_1$, yielding practically computable approximations with bounded errors. The approach is validated experimentally on synthetic manifolds (flat torus, sphere, Klein bottle) and hyperspheres, demonstrating robust discrimination between uniform and nonuniform samples and highlighting the method’s potential for unsupervised validation and uniformity assessment in manifold learning workflows.

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.

Figures (6)

• Figure 1: Constructing 3D embeddings of a Klein bottle lifted in ten dimensions, with varying success. From left to right, an output from PCA, ISOMAP, t-SNE, LLE, and spectral embedding. From visual inspection, PCA appears to have performed best in this example, followed by ISOMAP.
• Figure 2: Two realizations of a Binomial point processes. Left shows a homogeneous point process and right shows an inhomogeneous point process.
• Figure 3: From left to right: (1) uniform sample vs. (2) minor noise vs (3) stratification on the Klein bottle with $|X|=2000$. The right plot compares the normalized manifold density functions of each, with (1) in blue, (2) in orange, (3) red, and the theoretical manifold density function in green.
• Figure 4: From left to right: (1) uniform sample vs. (2) a stratification on the sphere, with $|X|=1000$. The right plot compares the manifold density functions of each, with (1) scaled in blue, (1) unscaled in red, (2) in orange, and the theoretical manifold density function in green.
• Figure 5: Embeddings of the Klein bottle in $\mathbb{R}^3$ after being projected from $\mathbb{R}^{10}$ from left to right: PCA, ISOMAP, and t-SNE (top) locally linear embedding and spectral embedding (bottom).

Theorems & Definitions (48)

• Definition 2.1: Uniformly Sampled Manifold Representation
• Theorem 2.1: Relation Between Volume and Curvature
• Theorem 2.2: Gauss-Bonnetlee2019intro
• Theorem 2.3: Gauss-Codazzi Equations for Hypersurfaces haizhong1996
• Example 2.1: Estimating Areas
• Definition 2.2: Manifold Learning
• Definition 3.1: Manifold Density Function And Its Estimators
• Definition 3.2: Manifold Score
• Theorem 3.1
• proof