Statistical Analysis of 2D Projective Shape Data Using a Novel Infimum Dimension Nash Embedding

A statistical analysis of manifold-valued data often requires an embedding into an Euclidean space of some kind. A good embedding will retain enough information to solve the problem at hand. In this regard, an isometric or Nash embedding is often desired since it preserves path lengths,angles, areas and volumes. Furthermore, we would also want an embedding to be equivariant so that it preserves geometric features associated with the natural homogeneity of the manifold. The Veronese-Whitney (VW) embedding is an equivariant embedding, in fact the only embedding, used in the analysis of 2D projective shape data, at the present time. In this manuscript we consider a novel equivariant Nash embedding into a Euclidean space of a lower dimension than that for the Veronese-Whitney embedding. In fact, it is shown in Patrangenaru and Paige (2023) that the novel equivariant Nash embedding places projective shape data in a Euclidean space with lowest possible dimension. We compare the performance of novel Nash embedding-based statistical techniques with those based on the VW embedding