  Given a set of points that sample a shape, the Rips complex of the data
points is often used in machine-learning to provide an approximation of the
shape easily-computed. It has been proved recently that the Rips complex
captures the homotopy type of the shape assuming the vertices of the complex
meet some mild sampling conditions. Unfortunately, the Rips complex is
generally high-dimensional. To remedy this problem, it is tempting to simplify
it through a sequence of collapses. Ideally, we would like to end up with a
triangulation of the shape. Experiments suggest that, as we simplify the
complex by iteratively collapsing faces, it should indeed be possible to avoid
entering a dead end such as the famous Bing's house with two rooms. This paper
provides a theoretical justification for this empirical observation.
  We demonstrate that the Rips complex of a point-cloud (for a well-chosen
scale parameter) can always be turned into a simplicial complex homeomorphic to
the shape by a sequence of collapses, assuming the shape is nicely triangulable
and well-sampled (two concepts we will explain in the paper). To establish our
result, we rely on a recent work which gives conditions under which the Rips
complex can be converted into a Cech complex by a sequence of collapses. We
proceed in two phases. Starting from the Cech complex, we first produce a
sequence of collapses that arrives to the Cech complex, restricted by the
shape. We then apply a sequence of collapses that transforms the result into
the nerve of some robust covering of the shape.
