Establish pushforward formulas for curvature estimation in arbitrary codimension (Grassmannian-valued tangent space noise)

Develop explicit transformation (pushforward) formulas for curvature estimation in the general case of submanifolds of arbitrary codimension, where tangent space estimates are modeled as Grassmannian-valued random variables with matrix von Mises–Fisher noise. Specifically, derive the analogs of Theorems 1 and 2 (noise pushforward for angle and curvature computations) for distributions on the Grassmannian Gr(m,n) under matrix vMF models, enabling decoded curvature estimation beyond codimension-one embeddings.

Background

The paper provides explicit pushforward distributions for angle and absolute variation curvature under von Mises–Fisher noise in the codimension-one setting, where tangent spaces are represented by unit normals on the sphere. Extending this approach to higher codimensions requires modeling tangent spaces as points on the Grassmannian with associated matrix-valued probability distributions.

The authors note that fully generalizing their probabilistic decoding framework to arbitrary codimension hinges on establishing explicit pushforward transformations from noisy tangent space estimates to curvature-related quantities, which remains to be completed.