Orientability and diffusion maps

One of the main objectives in the analysis of a high dimensional large data set is to learn its geometric and topological structure. Even though the data itself is parameterized as a point cloud in a high dimensional ambient space ^Rp, the correlation between parameters often suggests the "manifold assumption" that the data points are distributed on (or near) a low dimensional Riemannian manifold ^Md embedded in ^Rp, with d≪p. We introduce an algorithm that determines the orientability of the intrinsic manifold given a sufficiently large number of sampled data points. If the manifold is orientable, then our algorithm also provides an alternative procedure for computing the eigenfunctions of the Laplacian that are important in the diffusion map framework for reducing the dimensionality of the data. If the manifold is non-orientable, then we provide a modified diffusion mapping of its orientable double covering.