A robust distance measure for similarity-based classification on the SPD manifold

The symmetric positive definite (SPD) matrices, forming a Riemannian manifold, are commonly used as visual representations. The non-Euclidean geometry of the manifold often makes developing learning algorithms (e.g., classifiers) difficult and complicated. The concept of similarity-based learning has been shown to be effective to address various problems on SPD manifolds. This is mainly because the similarity-based algorithms are agnostic to the geometry and purely work based on the notion of similarities/distances. However, existing similarity-based models on SPD manifolds opt for holistic representations, ignoring characteristics of information captured by SPD matrices. To circumvent this limitation, we propose a novel SPD distance measure for the similarity-based algorithm. Specifically, we introduce the concept of point-to-set transformation, which enables us to learn multiple lower dimensional and discriminative SPD manifolds from a higher dimensional one. For lower dimensional SPD manifolds obtained by the point-to-set transformation, we propose a tailored set-to-set distance measure by making use of the family of alpha-beta divergences. We further propose to learn the point-to-set transformation and the set-to-set distance measure jointly, yielding a powerful similarity-based algorithm on SPD manifolds. Our thorough evaluations on several visual recognition tasks (e.g., action classification and face recognition) suggest that our algorithm comfortably outperforms various state-of-the-art algorithms.