Manifold learning using Euclidean k-nearest neighbor graphs [image processing examples

Abstract

In the manifold learning problem one seeks to discover a smooth low dimensional surface, i.e., a manifold embedded in a higher dimensional linear vector space, based on a set of n measured sample points on the surface. In this paper, we consider the closely related problem of estimating the manifold's intrinsic dimension and the intrinsic entropy of the sample points. Specifically, we view the sample points as realizations of an unknown multivariate density supported on an unknown smooth manifold. In previous work, we introduced a geometric probability method called the geodesic minimal spanning tree (GMST) to obtain asymptotically consistent estimates of manifold dimension and entropy. In this paper, we present a simpler method, based on the k-nearest neighbor (k-NN) graph that does not require estimation of geodesic distances on the manifold. The algorithm is applied to standard synthetic manifolds as well as real data sets consisting of images of faces.