A Riemannian geometric framework for manifold learning of non-Euclidean data

Abstract

A growing number of problems in data analysis and classification involve data that are non-Euclidean. For such problems, a naive application of vector space analysis algorithms will produce results that depend on the choice of local coordinates used to parametrize the data. At the same time, many data analysis and classification problems eventually reduce to an optimization, in which the criteria being minimized can be interpreted as the distortion associated with a mapping between two curved spaces. Exploiting this distortion minimizing perspective, we first show that manifold learning problems involving non-Euclidean data can be naturally framed as seeking a mapping between two Riemannian manifolds that is closest to being an isometry. A family of coordinate-invariant first-order distortion measures is then proposed that measure the proximity of the mapping to an isometry, and applied to manifold learning for non-Euclidean data sets. Case studies ranging from synthetic data to human mass-shape data demonstrate the many performance advantages of our Riemannian distortion minimization framework.