Geometry-Aware Similarity Learning on SPD Manifolds for Visual Recognition

Symmetric positive definite (SPD) matrices have been employed for data representation in many visual recognition tasks. The success is mainly attributed to learning discriminative SPD matrices encoding the Riemannian geometry of the underlying SPD manifolds. In this paper, we propose a geometry-aware SPD similarity learning (SPDSL) framework to learn discriminative SPD features by directly pursuing a manifold-manifold transformation matrix of full column rank. Specifically, by exploiting the Riemannian geometry of the manifolds of fixed-rank positive semidefinite (PSD) matrices, we present a new solution to reduce optimization over the space of column full-rank transformation matrices to optimization on the PSD manifold, which has a well-established Riemannian structure. Under this solution, we exploit a new supervised SPDSL technique to learn the manifold–manifold transformation by regressing the similarities of selected SPD data pairs to their ground-truth similarities on the target SPD manifold. To optimize the proposed objective function, we further derive an optimization algorithm on the PSD manifold. Evaluations on three visual classification tasks show the advantages of the proposed approach over the existing SPD-based discriminant learning methods.