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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.

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