Shift-invariant similarities circumvent distance concentration in stochastic neighbor embedding and variants

Dimensionality reduction aims at representing high-dimensional data in low-dimensional spaces, mainly for visualization and exploratory purposes. As an alternative to projections on linear subspaces, nonlinear dimensionality reduction, also known as manifold learning, can provide data representations that preserve structural properties such as pairwise distances or local neighborhoods. Very recently, similarity preservation emerged as a new paradigm for dimensionality reduction, with methods such as stochastic neighbor embedding and its variants. Experimentally, these methods significantly outperform the more classical methods based on distance or transformed distance preservation. This paper explains both theoretically and experimentally the reasons for these performances. In particular, it details why the phenonomenon of distance concentration is an impediment towards effcient dimensionality reduction and how SNE and its variants circumvent this diffculty by using similarities that are invariant to shifts with respect to squared distances. The paper also proposes a generalized definition of shift-invariant similarities that extend the applicability of SNE to noisy data.