Consistent Manifold Representation for Topological Data Analysis

TL;DR

This paper introduces the Continuous k-Nearest Neighbors (CkNN) graph construction, which consistently captures the topology and geometry of data sampled from manifolds, enabling improved topological data analysis and applications like clustering and pattern recognition.

Contribution

It proves the geometric and topological consistency of CkNN, establishing it as a unique unweighted graph construction that reflects the manifold's true structure in large data limits.

Findings

CkNN unnormalized graph Laplacian converges to the Laplace-Beltrami operator.

CkNN captures all topological features simultaneously.

Introduces a fast clustering algorithm based on CkNN.

Abstract

For data sampled from an arbitrary density on a manifold embedded in Euclidean space, the Continuous k-Nearest Neighbors (CkNN) graph construction is introduced. It is shown that CkNN is geometrically consistent in the sense that under certain conditions, the unnormalized graph Laplacian converges to the Laplace-Beltrami operator, spectrally as well as pointwise. It is proved for compact (and conjectured for noncompact) manifolds that CkNN is the unique unweighted construction that yields a geometry consistent with the connected components of the underlying manifold in the limit of large data. Thus CkNN produces a single graph that captures all topological features simultaneously, in contrast to persistent homology, which represents each homology generator at a separate scale. As applications we derive a new fast clustering algorithm and a method to identify patterns in natural images topologically. Finally, we conjecture that CkNN is topologically consistent, meaning that the homology of the Vietoris-Rips complex (implied by the graph Laplacian) converges to the homology of the underlying manifold (implied by the Laplace-de Rham operators) in the limit of large data.