Non-asymptotic Confidence Sets for Extrinsic Means on Spheres and Projective Spaces

TL;DR

This paper develops non-asymptotic confidence sets for extrinsic means on spheres, projective spaces, and Grassmann manifolds, providing rate-optimal guarantees and practical applications in projective shape analysis.

Contribution

It introduces a new method for constructing finite-sample confidence sets for extrinsic means on complex geometric spaces, extending previous asymptotic approaches.

Findings

Confidence sets are projections of Euclidean balls around sample means.

The confidence sets are rate-optimal under sufficient data concentration.

Application demonstrated in projective shape data analysis.

Abstract

Confidence sets from i.i.d. data are constructed for the extrinsic mean of a probabilty measure P on spheres, real projective spaces, and complex projective spaces, as well as Grassmann manifolds, with the latter three embedded by the Veronese-Whitney embedding. When the data are sufficiently concentrated, these are projections of a ball around the corresponding Euclidean sample mean. Furthermore, these confidence sets are rate-optimal. The usefulness of this approach is illustrated for projective shape data.