Dictionary Learning on Grassmann Manifolds

Abstract

Sparse representations have recently led to notable results in various visual recognition tasks. In a separate line of research, Riemannian manifolds have been shown useful for dealing with features and models that do not lie in Euclidean spaces. With the aim of building a bridge between the two realms, we address the problem of sparse coding and dictionary learning in Grassmann manifolds, i.e, the space of linear subspaces. To this end, we introduce algorithms for sparse coding and dictionary learning by embedding Grassmann manifolds into the space of symmetric matrices. Furthermore, to handle nonlinearity in data, we propose positive definite kernels on Grassmann manifolds and make use of them to perform coding and dictionary learning.