  Although Deep Learning (DL) has achieved success in complex Artificial
Intelligence (AI) tasks, it suffers from various notorious problems (e.g.,
feature redundancy, and vanishing or exploding gradients), since updating
parameters in Euclidean space cannot fully exploit the geometric structure of
the solution space. As a promising alternative solution, Riemannian-based DL
uses geometric optimization to update parameters on Riemannian manifolds and
can leverage the underlying geometric information. Accordingly, this article
presents a comprehensive survey of applying geometric optimization in DL. At
first, this article introduces the basic procedure of the geometric
optimization, including various geometric optimizers and some concepts of
Riemannian manifold. Subsequently, this article investigates the application of
geometric optimization in different DL networks in various AI tasks, e.g.,
convolution neural network, recurrent neural network, transfer learning, and
optimal transport. Additionally, typical public toolboxes that implement
optimization on manifold are also discussed. Finally, this article makes a
performance comparison between different deep geometric optimization methods
under image recognition scenarios.
