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Continuation of point clouds via persistence diagrams. (English) Zbl 1415.55006
Summary: 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.

##### MSC:
 55U10 Simplicial sets and complexes in algebraic topology 55N35 Other homology theories in algebraic topology 65H10 Numerical computation of solutions to systems of equations
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