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arXiv · 1506.03147

Continuation of Point Clouds via Persistence Diagrams

Marcio Gameiro·Yasuaki Hiraoka·Ippei Obayashi
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Abstract

In this paper, we present a mathematical and algorithmic framework for the continuation of point clouds by persistence diagrams. A key property used in the method is that the persistence map, which assigns a persistence diagram to a point cloud, is differentiable. This allows us to apply the Newton-Raphson continuation method in this setting. Given an original point cloud $P$, its persistence diagram $D$, and a target persistence diagram $D'$, we gradually move from $D$ to $D'$, by successively computing intermediate point clouds until we finally find a point cloud $P'$ having $D'$ as its persistence diagram. Our method can be applied to a wide variety of situations in topological data analysis where it is necessary to solve an inverse problem, from persistence diagrams to point cloud data.

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Marcio Gameiro, Yasuaki Hiraoka, Ippei Obayashi. 2015-06-10. Continuation of Point Clouds via Persistence Diagrams. https://doi.org/10.1016/j.physd.2015.11.011

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