  Appropriately representing elements in a database so that queries may be
accurately matched is a central task in information retrieval. This recently
has been achieved by embedding the graphical structure of the database into a
manifold so that the hierarchy is preserved. Persistent homology provides a
rigorous characterization for the database topology in terms of both its
hierarchy and connectivity structure. We compute persistent homology on a
variety of datasets and show that some commonly used embeddings fail to
preserve the connectivity. Moreover, we show that embeddings which successfully
retain the database topology coincide in persistent homology. We introduce the
dilation-invariant bottleneck distance to capture this effect, which addresses
metric distortion on manifolds. We use it to show that distances between
topology-preserving embeddings of databases are small.
