Riemannian Optimization

Riemannian optimization addresses the challenge of optimizing functions whose variables are constrained to lie on a curved surface, or manifold, rather than in a flat Euclidean space. Current research focuses on developing efficient algorithms, such as Riemannian gradient descent, coordinate descent, and Newton-type methods, often incorporating variance reduction and acceleration techniques, for various manifolds including Stiefel, Grassmann, and SPD manifolds. These advancements are impacting diverse fields, improving performance in machine learning (e.g., deep learning, tensor factorization), computer vision (e.g., pose graph optimization, registration), and quantum computing (e.g., quantum process tomography), by enabling the efficient solution of complex, constrained optimization problems.