Riemannian Stochastic

Riemannian stochastic optimization adapts stochastic gradient descent methods to optimize functions defined on curved spaces (Riemannian manifolds), addressing challenges posed by data residing in non-Euclidean domains. Current research focuses on developing and analyzing algorithms like Riemannian stochastic gradient descent (RSGD) and its variants, including federated learning adaptations and zeroth-order methods, often employing retractions to improve computational efficiency. This field is significant because it enables the application of powerful optimization techniques to problems in machine learning (e.g., Grassmann manifold representations for improved accuracy and transferability) and other areas where data naturally resides on manifolds, leading to more accurate and efficient solutions.