Dimensionality Reduction has Quantifiable Imperfections: Two Geometric Bounds Gavin Weiguang Ding Borealis AI Borealis AI Canada

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.