Towards Morse Theory for Point Cloud Data

Abstract : Morse theory provides a powerful framework to study the topology of a manifold from a function defined on it, but discrete constructions have remained elusive due to the difficulty of translating smooth concepts to the discrete setting. Consider the problem of approximating the Morse-Smale (MS) complex of a Morse function from a point cloud and an associated nearest neighbor graph (NNG). While following the constructive proof of the Morse homology theorem, we present novel concepts for critical points of any index, and the associated Morse-Smale diagram. Our framework has three key advantages. First, it requires elementary data structures and operations, and is thus suitable for high-dimensional data processing. Second, it is gradient free, which makes it suitable to investigate functions whose gradient is unknown or expensive to compute. Third, in case of under-sampling and even if the exact (unknown) MS diagram is not found, the output conveys information in terms of ambiguous flow, and the Morse theoretical version of topological persistence, which consists in canceling critical points by flow reversal, applies.\\ %% On the experimental side, we present a comprehensive analysis of a large panel of bi-variate and tri-variate Morse functions whose Morse-Smale diagrams are known perfectly, and show that these diagrams are recovered perfectly. \smallskip In a broader perspective, we see our framework as a first step to study complex dynamical systems from mere samplings consisting of point clouds.
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[Research Report] RR-8331, INRIA. 2013, pp.37
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Frédéric Cazals, Andrea Roth, Charles Robert, Mueller Christian. Towards Morse Theory for Point Cloud Data. [Research Report] RR-8331, INRIA. 2013, pp.37. 〈hal-00848753〉

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