Estimating Riemannian Quantities¶
Overview¶
This project studies how to recover the Riemannian geometric structure of a submanifold from noisy observations in Euclidean space. Our main result is that, under a Gaussian corruption model, several fundamental Riemannian quantities, including tangent spaces, intrinsic dimension, and the second fundamental form, are identifiable from derivatives of the noisy density.
Our approach is built on the small-noise structure of the noisy density. We show that the log-density behaves like a scaled squared distance to the manifold, and its first three derivatives encode, respectively, a normal field, a tangent-normal spectral splitting, and the variation of the tangential projection. These structures lead to population-level geometric estimators and sample-level plug-in estimators based on kernel density derivatives.
Furthermore, we connect these constructions to density-induced ambient metrics, providing a geometric interpretation of how the underlying manifold gives rise to an ambient geometric structure through the noisy density.
Our approach is built on the small-noise structure of the noisy density. We show that the log-density behaves like a scaled squared distance to the manifold, and its first three derivatives encode, respectively, a normal field, a tangent-normal spectral splitting, and the variation of the tangential projection. These structures lead to population-level geometric estimators and sample-level plug-in estimators based on kernel density derivatives.
Furthermore, we connect these constructions to density-induced ambient metrics, providing a geometric interpretation of how the underlying manifold gives rise to an ambient geometric structure through the noisy density.
Detailed description and discussion can be found in paper:
To cite: