Entropic graphs for intrinsic dimension estimation in manifold learning

Many interesting data sets, although high dimensional in nature, can be characterized by a low dimensional non linear parameterization, if, for example, the data set lies on a manifold. In this paper we consider the problem of estimating the manifold´s intrinsic dimension and the intrinsic entropy of the data set. Specifically, we view the data as realizations of an unknown multivariate density supported on an unknown smooth manifold. We present a novel geometric approach, based on asymptotic properties of entropic graphs, to obtain asymptotically consistent estimates of the manifold dimension and the Renyi α-entropy of the data density on the manifold. The proposed algorithm simply constructs a minimal spanning tree (MST) sequence using a geodesic distance matrix and uses the overall lengths of the MSTs to compute the desired estimators. We apply the algorithm to standard synthetic manifolds as well as to real data sets.