  In this paper, we investigate Dimensionality reduction (DR) maps in an
information retrieval setting from a quantitative topology point of view. In
particular, we show that no DR maps can achieve perfect precision and perfect
recall simultaneously. Thus a continuous DR map must have imperfect precision.
We further prove an upper bound on the precision of Lipschitz continuous DR
maps. While precision is a natural measure in an information retrieval setting,
it does not measure `how' wrong the retrieved data is. We therefore propose a
new measure based on Wasserstein distance that comes with similar theoretical
guarantee. A key technical step in our proofs is a particular optimization
problem of the $L_2$-Wasserstein distance over a constrained set of
distributions. We provide a complete solution to this optimization problem,
which can be of independent interest on the technical side.
