Manifold learning using geodesic entropic graphs

Summary form only given. 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 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. We present a novel geometrical probability approach, called the geodesic entropic graph (GET) method, to obtaining asymptotically consistent estimates of the manifold dimension and the Renyi α-entropy of the sample density on the manifold. The GET approach is striking in its simplicity and does not require reconstructing the manifold or estimating the multivariate density of the samples. The GET method simply constructs an entropic graph, e.g., a minimal spanning tree (MST) or k-nearest neighbor graph (k-NNG), to estimate the geodesic neighborhoods connecting points on the manifold. The growth rate of the length functional of the entropic graph is then used to simultaneously estimate manifold dimension and sample entropy.