Skip to yearly menu bar Skip to main content


Dimensionality Reduction has Quantifiable Imperfections: Two Geometric Bounds

Yik Chau (Kry) Lui · Gavin Weiguang Ding · Ruitong Huang · Robert McCann

Room 517 AB #103

Keywords: [ Learning Theory ] [ Nonlinear Dimensionality Reduction and Manifold Learning ]

Abstract: 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.

Live content is unavailable. Log in and register to view live content