  Persistent homology of the Rips filtration allows to track topological
features of a point cloud over scales, and is a foundational tool of
topological data analysis. Unfortunately, the Rips-filtration is exponentially
sized, when considered as a filtered simplicial complex. Hence, the computation
of full persistence modules is impossible for all but the tiniest of datasets;
when truncating the dimension of topological features, the situation becomes
slightly less intractable, but still daunting for medium-sized datasets.
  It is theoretically possible to approximate the Rips-filtration by a much
smaller and sparser, linear-sized simplicial complexs, however, possibly due to
the complexity of existing approaches, we are not aware of any existing
implementation.
  We propose a different sparsification scheme, based on cover-trees, that is
easy to implement, while giving similar guarantees on the computational
scaling. We further propose a visualization that is adapted to approximate
persistence diagrams, by incorporating a variant of error bars and keeping
track of all approximation guarantees, explicitly.
