 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  .
In topological data analysis, persistent homology is used to study the "shape of data". 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 prove 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 the average persistence landscape, a continuous mapping from metric spaces with a probability measure to a Hilbert space . In the present application, the average persistence landscapes of points sampled from disks of constant curvature results in a path in this Hilbert space which may be learned using standard tools from statistical and machine learning .
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In topological data analysis, persistent homology is used to study the "shape of data".
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show
prove
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average persistence landscapes
the average persistence landscape, a continuous mapping from metric spaces with a probability measure to a Hilbert space
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produce
results in
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a
this
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using standard tools from statistical and machine learning