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TOPOLOGICAL CONSISTENCY VIA KERNEL ESTIMATION
, 2014
"... We introduce a consistent estimator for the homology (an algebraic structure representing connected components and cycles) of superlevel sets of both density and regression functions. Our method is based on kernel estimation. We apply this procedure to two problems: 1) inferring the homology struc ..."
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We introduce a consistent estimator for the homology (an algebraic structure representing connected components and cycles) of superlevel sets of both density and regression functions. Our method is based on kernel estimation. We apply this procedure to two problems: 1) inferring the homology structure of manifolds from noisy observations, 2) inferring the persistent homology (a multiscale extension of homology) of either density or regression functions. We prove consistency for both of these problems. In addition to the theoretical results we demonstrate these methods on simulated data for binary regression and clustering applications.
TOPOLOGY OF RANDOM GEOMETRIC COMPLEXES: A SURVEY
"... In this expository article, we survey the rapidly emerging area of random geometric simplicial complexes. Random simplicial complexes may be viewed as higherdimensional generalizations of random graphs. Perhaps the most studied model of random graph is the Erdős–Rényi model G(n, p), where every ..."
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In this expository article, we survey the rapidly emerging area of random geometric simplicial complexes. Random simplicial complexes may be viewed as higherdimensional generalizations of random graphs. Perhaps the most studied model of random graph is the Erdős–Rényi model G(n, p), where every edge appears inde