Empirical geodesic graphs and CAT(k) metrics for data analysis

A methodology is developed for data analysis based on empirically constructed geodesic metric spaces. The population version defines distance by the amount of probability mass accumulated on traveling between two points and geodesic metric arises from the shortest path version. Such metrics are then transformed in a number of ways to produce families of geodesic metric spaces. Empirical versions of the geodesics allow computation of intrinsic means and associated measures of dispersion. A version of the empirical geodesic is introduced based on the graph of some initial complex such as the Delaunay complex. For certain parameter ranges the spaces become CAT(0) spaces and the intrinsic means are unique. In the graph case a minimal spanning tree obtained as a limiting case is CAT(0). In other cases the aggregate squared distance from a test point provides local minima which yield information about clusters. This is particularly relevant for metrics based on so-called metric cones which allow extensions to CAT(k) spaces. After gaining some theoretical understanding the methods are tested on small artificial examples.