A Novel Infimum Dimension Nash Embedding for 3D Projective Shape Data

An isometric embedding of a Riemannian manifold in an Euclidean space is an embedding suchthat the Riemannian structure on each tangent space is induced by the scalar product in the ambientspace; such an embedding preserves path lengths, angles, areas and volumes. Such a mappingpreserves much of the geometric information contained in the original Riemannian manifold wherethe data is valued allowing for some computationally advantages. In addition if the Riemmanianmanifold is homogeneous then we would also want the such an embedding, to be equivariant. TheVeronese-Whitney (VW) embedding is an isometric equivariant embedding for 3D projective shapedata. It is in fact the only embedding used in the analysis of 3D projective shape data, at the presenttime. In this paper we consider a novel equivariant isometric embedding, the Nash embeddinginto a Euclidean space of a lower dimension than the Euclidean space where the Veronese-Whitneyembedding is valued . We compare the performance of the novel Nash embedding-based statisticaltechniques with those based on the VW embedding in Monte Carlo studies and a real data example.