High-dimensional labeled data analysis with topology representing graphs

We propose the use of topology representing graphs for the exploratory analysis of high-dimensional labeled data. The Delaunay graph contains all the topological information needed to analyze the topology of the classes (e.g. the number of separate clusters of a given class, the way these clusters are in contact with each other or the shape of these clusters). The Delaunay graph also allows to sample the decision boundary of the Nearest Neighbor rule, to define a topological criterion of non-linear separability of the classes and to find data which are near the decision boundary so that their label must be considered carefully. This graph then provides a way to analyze the complexity of a classification problem, and tools for decision support. When the Delaunay graph is not tractable in too high-dimensional spaces, we propose to use the Gabriel graph instead and discuss the limits of this approach. This analysis technique is complementary with projection techniques, as it allows to handle the data as they are in the data space, avoiding projection distortions. We apply it to analyze the well-known Iris database and a seismic events database.