How can one determine the curvature of data and how does it help to derive the salient structural features of a data set?

After determining the appropriate model to represent data, the next step is to derive the salient structural features of data based on the tools available in that specific mathematical model. While in topological data analysis the objective is to extract quantitative features, the shape of data, geometric data analysis mainly deals with quantitative features of data. For instance, the prominent scheme of manifold learning is applied to find the comparatively low dimensional Riemannian manifold on which the data set fits best. It then raises the question of whether one can anticipate some geometric properties from initial model before finding this manifold structure.

The most important quantitative measures that in a good extent reveal the geometry of a Riemannian manifold are its (sectional and Ricci) curvatures. Although, these quantities were originally defined infinitesimally through certain combinations of second and first derivatives of the Riemannian metric tensor, there is a way to define either these quantities themselves or a (lower or upper) bound for them that no longer need derivatives for their evaluation and is therefore generalizable to metric spaces.

The aim of this presentation is to briefly present these generalized curvatures and see which kind of relations between data points each of them evaluate and what kind of information they reveal.