Manifold stability and the central limit theorem for mean shape

Abstract

In the literarature there is an overwhelming number of concepts for a “mean” on shape spaces. First we classify those into three fundamental types. Then we introduce manifold stability which is essential to a apply a standard two sample test on a non-manifold shape space. For sample means on Kendall’s shape spaces, we give the proof to establish manifold stability and illustrate consequences for the discrimination of non-concentrated shapes. We conclude with a discussion of the result for general shape spaces and population means.