Infimum Dimension Nash Embeddings for 2D Projective Shape Analysis

Abstract

Vector embeddings make complicated data extracted from networks, words and images, more amendable to data science applications. At the present time, the Veronese-Whitney (VW) matrix embedding of the real projective space is the state of the art for making inference about digital images from an uncalibrated camera, such as a cell phone or security camera. In this work we consider vector embeddings for the projective shape data and in particular determine the minimum dimension isometric (distance-preserving or Nash) vector embedding for a projective space. We determine such an embedding for the projective plane in closed-form. From this embedding we determine an embedding for the Cartesian product of projective planes which is used to develop a novel extrinsic mean test as well as a novel homogeneity test for 2D projective shape analysis. In a Monte Carlo study and real data application it is found that this new testing procedure performs as well as the state of the art extrinsic test based on the VW embedding in terms of hypothesis tests for extrinsic means, tests for homogeneity with tangential components and classification via support vector machines. Furthermore, it generally outperforms the vech of the VW embedding. Note however that the Nash embedding is into five-dimensional Euclidean space, whereas the VW embedding is into the Euclidean space of 3 by 3 symmetric matrices, which is six-dimensional. Our vector-valued Nash embedding is preferred over the matrix-valued VW embedding for data science applications since (i) it is a vector embedding and performs as well as the state of the art VW matrix embedding when the latter can be used in a statistical procedure and (ii) and our embedding is easily used for classification and visualization with traditional statistical techniques.