Consistent Manifold Representation for Topological Data Analysis

For data sampled from an arbitrary density on a manifold embedded in Euclidean space, we introduce the Continuous k-Nearest Neighbors (CkNN) graph construction. We prove that CkNN is the unique unweighted construction that is consistent with the underlying manifold topology in the limit of large data, for compact Riemannian manifolds and a large class of non-compact manifolds. In contrast to persistent homology, which represents each homology generator at a separate scale, CkNN produces a single graph that captures all topological features simultaneously. This requires new results on the spectral convergence of the graph Laplacian as a consistent estimator of the Laplacede Rham operator. As applications we derive a new fast clustering algorithm and a method to identify patterns in natural images topologically.