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 connected components of the underlying manifold in the limit of large data, for compact Riemannian manifolds and a large class of non-compact manifolds. More precisely, we show that CkNN is geometrically consistent in the sense that the unnormalized graph Laplacian converges to the Laplace-Beltrami operator, spectrally as well as pointwise. We demonstrate that 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.