Nonlinear data projection on non-Euclidean manifolds with controlled trade-off between trustworthiness and continuity

Abstract

This paper presents a framework for nonlinear dimensionality reduction methods aimed at projecting data on a non-Euclidean manifold, when their structure is too complex to be embedded in an Euclidean space. The methodology proposes an optimization procedure on manifolds to minimize a pairwise distance criterion that implements a control of the trade-off between trustworthiness and continuity, two criteria that, respectively, represent the risks of flattening and tearing the projection. The methodology is presented as general as possible and is illustrated in the specific case of the sphere.