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

Cheongjae Jang, Yung-Kyun Noh, Frank C. Park

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 nonEuclidean data sets. Case studies ranging from synthetic data to human mass-shape
data demonstrate the many performance advantages of our Riemannian distortion
minimization framework.