Dictionary Learning on Riemannian Manifolds

Existing dictionary learning algorithms rely heavily on the assumption that the data points are vectors in some Euclidean space R, and the dictionary is learned from the input data using only the vector space structure of R. However, in many applications, features and data points often belong to some Riemannian manifold with its intrinsic metric structure that is potentially important and critical to the application. The extrinsic viewpoint of existing dictionary learning methods becomes inappropriate and inadequate if the intrinsic geometric structure is required to be incorporated in the model. In this paper, we develop a very general dictionary learning framework for data lying on some known Riemannnian manifolds. Using the local linear structures furnished by the Riemannian geometry, we propose a novel dictionary learning algorithm that can be considered as data-specific, a feature that is not present in the existing methods. We show that both the dictionary and sparse coding can be effectively computed for Riemannian manifolds. We validate the proposed method using a classification problem in medical image analysis. The preliminary results demonstrate that the dictionary learned using the proposed method can and does indeed provide real improvements when compared with other direct approaches.