Persistent vs. Consistent Homology from Point Clouds

Tim Sauer (September 14, 2016)

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Persistent homology has become a standard tool in modern data analysis, identifying homology generators at various scales as a parameter representing distances between points is increased. We show that for general sampling from a Riemannian manifold, there is a graph construction that captures all topological features in a single graph, which we call `consistent' homology. More precisely, the graph converges spectrally to the Laplace-De Rham operator of the manifold in the limit of large data. The graph construction, called continuous k-nearest neighbors (CkNN), neutralizes nonuniform sampling, and in practice reduces data requirements as well. We examine under which circumstances persistent or consistent approaches are preferred, and illustrate with data from neural cultures and physical experiments.