Essentials of backward nested descriptors inference

Abstract

Principal component analysis (PCA) is a popular device for dimension reduction and their asymptotics are well known. In particular, principal components through the mean span the data with decreasing residual variance, as the dimension increases, or, equivalently maximize projected variance, as the dimensions decrease, and these spans are nested in a backward and forward fashion – all due to Pythagoras Theorem. For non-Euclidean data with no Pythagorian variance decomposition available, it is not obvious what should take the place of PCA and how asymptotic results generalize. For spaces with high symmetry, for instance for spheres, backward nested sphere analysis has been successfully introduced. For spaces with less symmetry, recently, nested barycentric subspaces have been proposed. In this short contribution we sketch how to arrive at asymptotic results for sequences of random nested subspaces.