  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.
