Curvature of high-dimensional data

Abstract

We consider the problem of estimating curvature where the data can be viewed as a noisy sample from an underlying manifold. For manifolds of dimension greater than one there are multiple definitions of local curvature, each suggesting a different estimation process for a given data set. Recently, there has been progress in proving that estimates of ``local point cloud curvature" converge to the related smooth notion of local curvature as the density of the point cloud approaches infinity. Herein we investigate practical limitations of such convergence theorems and discuss the significant impact of bias in such estimates as reported in recent literature. We provide theoretical arguments for the fact that bias increases drastically in higher dimensions, so much so that in high dimensions, the probability that a naive curvature estimate lies in a small interval near the true curvature could be near zero. We present a probabilistic framework that enables the construction of more accurate estimators of curvature for arbitrary noise models. The efficacy of our technique is supported with experiments on spheres of dimension as large as twelve.