Appropriately representing elements in a database so that queries may be accurately matched is a central task in information retrieval; recently, this has been achieved by embedding the graphical structure of the database into a manifold in a hierarchy-preserving manner using a variety of metrics. Persistent homology is a tool commonly used in topological data analysis that is able to rigorously characterize a database in terms of both its hierarchy and connectivity structure. Computing persistent homology on a variety of embedded datasets reveals that some commonly used embeddings fail to preserve the connectivity. We show that those embeddings which successfully retain the database topology coincide in persistent homology by introducing two dilation-invariant comparative measures to capture this effect: in particular, they address the issue of metric distortion on manifolds. We provide an algorithm for their computation that exhibits greatly reduced time complexity over existing methods. We use these measures to perform the first instance of topology-based information retrieval and demonstrate its increased performance over the standard bottleneck distance for persistent homology. We showcase our approach on databases of different data varieties including text, videos, and medical images.
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