Set-to-Set Distance Metric Learning on SPD Manifolds

Abstract

The Symmetric Positive Definite (SPD) matrix on the Riemannian manifold has become a prevalent representation in many computer vision tasks. However, learning a proper distance metric between two SPD matrices is still a challenging problem. Existing metric learning methods of SPD matrices only regard an SPD matrix as a global representation and thus ignore different roles of intrinsic properties in the SPD matrix. In this paper, we propose a novel SPD matrix metric learning method of discovering SPD matrix intrinsic properties and measuring the distance considering different roles of intrinsic properties. In particular, the intrinsic properties of an SPD matrix are discovered by projecting the SPD matrix to multiple low-dimensional SPD manifolds, and the obtained low-dimensional SPD matrices constitute a set. Accordingly, the metric between two original SPD matrices is transformed into a set-to-set metric on multiple low-dimensional SPD manifolds. Based on the learnable alpha-beta divergence, the set-to-set metric is computed by summarizing multiple alpha-beta divergences assigned on low-dimensional SPD manifolds, which models different roles of intrinsic properties. The experimental results on four visual tasks demonstrate that our method achieves the state-of-the art performance.