Graph Laplacians on Singular Manifolds: Toward understanding complex spaces: graph Laplacians on manifolds with singularities and boundaries

TL;DR

This paper investigates how graph Laplacians behave near boundaries and singularities in manifolds, revealing distinct scaling behaviors and implications for analyzing complex, realistic data structures.

Contribution

It provides a detailed analysis of graph Laplacian behavior at singularities and boundaries, highlighting differences from the interior and implications for data analysis.

Findings

Graph Laplacian near singularities tends to a first-order differential operator.

Scaling behavior near singularities differs from that in the interior of manifolds.

Singularities contribute disproportionately to the overall behavior despite occupying small volume.

Abstract

Recently, much of the existing work in manifold learning has been done under the assumption that the data is sampled from a manifold without boundaries and singularities or that the functions of interest are evaluated away from such points. At the same time, it can be argued that singularities and boundaries are an important aspect of the geometry of realistic data. In this paper we consider the behavior of graph Laplacians at points at or near boundaries and two main types of other singularities: intersections, where different manifolds come together and sharp "edges", where a manifold sharply changes direction. We show that the behavior of graph Laplacian near these singularities is quite different from that in the interior of the manifolds. In fact, a phenomenon somewhat reminiscent of the Gibbs effect in the analysis of Fourier series, can be observed in the behavior of graph Laplacian near such points. Unlike in the interior of the domain, where graph Laplacian converges to the Laplace-Beltrami operator, near singularities graph Laplacian tends to a first-order differential operator, which exhibits different scaling behavior as a function of the kernel width. One important implication is that while points near the singularities occupy only a small part of the total volume, the difference in scaling results in a disproportionately large contribution to the total behavior. Another significant finding is that while the scaling behavior of the operator is the same near different types of singularities, they are very distinct at a more refined level of analysis. We believe that a comprehensive understanding of these structures in addition to the standard case of a smooth manifold can take us a long way toward better methods for analysis of complex non-linear data and can lead to significant progress in algorithm design.