Continuation of point clouds via persistence diagrams
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 NewtonRaphson 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.
 Publication:

Physica D Nonlinear Phenomena
 Pub Date:
 November 2016
 DOI:
 10.1016/j.physd.2015.11.011
 arXiv:
 arXiv:1506.03147
 Bibcode:
 2016PhyD..334..118G
 Keywords:

 Point cloud;
 Persistent homology;
 Persistence diagram;
 Continuation;
 Mathematics  Numerical Analysis;
 Computer Science  Computational Geometry;
 Mathematics  Algebraic Topology;
 Mathematics  Dynamical Systems
 EPrint:
 doi:10.1016/j.physd.2015.11.011