Determining Intrinsic Dimension and Entropy of High-Dimensional Shape Spaces

摘要: Given a finite set of random samples from smooth Riemannian manifold embedded in ℝd, two important questions are: what is the intrinsic dimension and entropy underlying sampling distribution on manifold? These naturally arise study shape spaces generated by images or signals for purposes classification, compression, reconstruction. This chapter concerned with simple estimators based lengths geodesic minimal spanning tree (GMST) k-nearest neighbor (k-NN) graph. We provide proofs strong consistency these under weak assumptions compactness boundedness Lebesgue density supported manifold. illustrate MNIST database handwritten digits.