Riemannian quasi-Newton methods, implementation techniques, and applications

LI Xudong, School of Data Science, Fudan University

577998

In the Euclidean setting, quasi-Newton methods have been widely used for solving many important problems. When considering a cost function defined on a Riemannian manifold, the Euclidean quasi-Newton methods cannot be applied directly and many generic Riemannian quasi-Newton methods have been proposed. In this presentation, the generic Riemannian BFGS and Riemannian SR1 methods are introduced. An important implementation technique is discussed. A limited-memory version of a line search Riemannian BFGS method and a limited-memory version of trust region Riemannian SR1 method are also given. Finally, numerical experiments are used to demonstrate the performance of Riemannian quasi-Newton methods.

HUANG Wen is an associate professor at Xiamen University. He finished his Ph.D in Applied and Computational Mathematics in 2014 at Florida State University. Dr. Huang's current interests are optimization on Riemannian manifolds and its applications, including elastic shape analysis, independent component analysis, synchronization of rotations, phase retrieval problem, blind deconvolution, computations of symmetric positive definite matrices, and role model problems; the theoretical and algorithmic aspects of large data-driven problems. He developed software, called TreeScaper, for phylogenetic analysis and a C++ toolbox, ROPTLIB, for Riemannian optimization.