Continuity vs. Injectivity in Dimensionality Reduction: a Quantitative Topology View

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 `howu0027 wrong the retrieved data is. We therefore propose a new measure based on Wasserstein distance that comes with theoretical guarantee in terms of a lower bound. A key technical step in our proofs is a particular optimization problem of the L2-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. Lastly, simulation confirms the tightness of the bound.