Exploring Singularities in point clouds with the graph Laplacian: An explicit approach

TL;DR

This work develops explicit, sharp bounds for the graph Laplacian $L_{n,t}$ acting on linear functionals $f(x)=v\cdot x$ near singularities of a data manifold $\Omega=\bigcup_i\Omega_i$, including intersections and boundaries. By decomposing the operator into restricted pieces $L_t^i$ on each component and deriving asymptotic forms such as $L_t^i f(x) = t^{d/2+1/2} A(d,r_0,\theta) v_{n,\Omega_i}\sin\theta\,r\,e^{ - \sin^2\theta\,r^2} + t^{d/2}B(x)e^{-r_0^2}$ (plus boundary-augmented terms in the non-flat case), the authors obtain finite-sample concentration bounds and construct hypothesis tests for singularities, as well as estimators for intersection location and angle. The results cover flat unions, general manifolds, and noisy samples, with numerical experiments validating tests on neural network loss-sets and demonstrating reliable estimation of geometric features in both flat and curved settings. The work advances practical tools for revealing and quantifying singular geometry in high-dimensional data, with potential applications to neural network landscapes and manifold-structured data analysis.

Abstract

We develop theory and methods that use the graph Laplacian to analyze the geometry of the underlying manifold of datasets. Our theory provides theoretical guarantees and explicit bounds on the functional forms of the graph Laplacian when it acts on functions defined close to singularities of the underlying manifold. We use these explicit bounds to develop tests for singularities and propose methods that can be used to estimate geometric properties of singularities in the datasets.

Figures (12)

• Figure 1: Graph Laplacian $L_{n,t}$ acting on a linear function $f$. Purple color showing positive, and green color negative values of $L_{n,t}f$, where lack of color indicates values near 0
• Figure 2: There is a singularity in the intersection of the lines above. The left figure shows a point of \ref{['it:type4']}, and the right figure shows a point of \ref{['it:type2']}.
• Figure 3: Schematic picture of the geometry of \ref{['thm:explicit:intersection']}, where $\Omega_1$ is the object of interest and $x \in \Omega_2$ for visualization purposes.
• Figure 4: Results of hypothesis test for singularities in flat manifolds, each experiment run 100 times.
• Figure 5: Two visualizations of neural network analysis: (left) PCA projection of a neural network's zero set from $\mathbb{R}^6$ to $\mathbb{R}^3$; (right) Graph Laplacian analysis showing $L_t f(x)$ on the $y$-axis and $f(x)$ on the $x$-axis, where $f(x) = v \cdot x$. The dashed line in the right plot represents the rejection region $\Theta$ specified in \ref{['thm:hypothesis']}.

Theorems & Definitions (22)

• Definition 3.1
• Example 3.2
• Remark 3.3
• Remark 3.4
• Theorem 1
• Theorem 2
• Theorem 3
• Remark 4.1
• Theorem 4: General manifold
• Lemma 4.2