Geometry aware constrained optimization techniques for deep learning

In this paper, we generalize the Stochastic Gradient Descent (SGD) and RMSProp algorithms to the setting of Riemannian optimization. SGD is a popular method for large scale optimization. In particular, it is widely used to train the weights of Deep Neural Networks. However, gradients computed using standard SGD can have large variance, which is detrimental for the convergence rate of the algorithm. Other methods such as RMSProp and ADAM address this issue. Nevertheless, these methods cannot be directly applied to constrained optimization problems. In this paper, we extend some popular optimization algorithm to the Riemannian (constrained) setting. We substantiate our proposed extensions with a range of relevant problems in machine learning such as incremental Principal Component Analysis, computating the Riemannian centroids of SPD matrices, and Deep Metric Learning. We achieve competitive results against the state of the art for fine-grained object recognition datasets.