The Riemannian Barzilai-borwein Method with Nonmonotone Line-search and the Matrix Geometric Mean Computation

The Barzilai-Borwein method, an effective gradient descent method with cleaver choice of the step-length, is adapted from nonlinear optimization to Riemannian manifold optimization. More generally, global convergence of a nonmonotone line-search strategy for Riemannian optimization algorithms is proved under some standard assumptions. By a set of numerical tests, the Riemannian Barzilai-Borwein method with nonmonotone line-search is shown to be competitive in several Riemannian optimization problems. When used to compute the matrix geometric mean, known as the Karcher mean of positive definite matrices, it notably outperforms existing first-order optimization methods. Riemannian optimization; manifold optimization; Barzilai-Borwein algorithm; nonmonotone line-search; Karcher mean; matrix geometric mean; positive definite matrix.