Learning tubes

We present a new method for analyzing data manifolds based on Weyl's tube theorem. The coefficients of the tube polynomial for a manifold provide geometric information such as the volume of the manifold or its Euler characteristic, thus providing bounds on the geometric nature of the manifold. We present an algorithm estimating the coefficients of the tube polynomial for a given manifold and demonstrate the features of our algorithm on artificial data sets. We apply the algorithm on a real-world traffic data set to determine the number and properties of clusters. We furthermore demonstrate that our algorithm can be used to determine image coverage of an object, giving hints on where a manifold is not sufficiently sampled.