Persistent homology detects curvature
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Persistent homology computations are completely characterized by a set of intervals called a bar code. It is often said that the long intervals represent the "topological signal" and the short intervals represent "noise". We give evidence to dispute this thesis, showing that the short intervals encode geometric information. Specifically, we show that persistent homology detects the curvature of disks from which points have been sampled. We describe a general computational framework for solving inverse problems using average persistence landscapes. In the present application, the average persistence landscapes of points sampled from disks of constant curvature produce a path in a Hilbert space which may be learned.
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