Assess representativeness of finite-density curvature estimates under noise

Determine whether curvature values computed from point clouds with finite sampling density, in the presence of ambient noise, reliably represent the true curvature of the underlying manifold. Formulate conditions or bounds under which finite-sample estimators—particularly those based on tangent space methods—yield unbiased or acceptably biased approximations to true curvature.

Background

The paper reviews recent literature showing convergence results for curvature estimators in the infinite-sample limit but highlights systematic bias in finite-sample, noisy settings, especially in higher dimensions.

The authors emphasize that despite progress, the practical reliability of curvature estimates at finite densities remains uncertain and is of central importance for manifold learning and data geometry applications.