Learning to Optimize on SPD Manifolds
Zhi Gao,  Yuwei Wu,  Yunde Jia,  Mehrtash Harandi

Many tasks in computer vision and machine learning are modeled as optimization problems with constraints in the form of Symmetric Positive Definite (SPD) matrices. Solving such optimization problems is challenging due to the non-linearity of the SPD manifold, making optimization with SPD constraints heavily relying on expert knowledge and human involvement. In this paper, we propose a meta-learning method to automatically learn an iterative optimizer on SPD manifolds. Specifically, we introduce a novel recurrent model that takes into account the structure of input gradients and identifies the updating scheme of optimization. We parameterize the optimizer by the recurrent model and utilize Riemannian operations to ensure that our method is faithful to the geometry of SPD manifolds. Compared with existing SPD optimizers, our optimizer effectively exploits the underlying data distribution and learns a better optimization trajectory in a data-driven manner. Extensive experiments on various computer vision tasks including metric nearness, clustering, and similarity learning demonstrate that our optimizer outperforms existing state-of-the-art methods consistently.