A Riemannian Derivative-Free Polak-Ribiere-Polyak Method for Tangent Vector Field
Teng-Teng Yao, Zhi Zhao, Zheng-Jian Bai, Xiao-Qing Jin
TL;DR
This work targets finding zeros of tangent vector fields $F$ on Riemannian manifolds by reframing the problem as the unconstrained minimization of $f(X)=\tfrac{1}{2}\|F(X)\|^2$ and introducing a Riemannian derivative-free Polak–Ribiére–Polyak (PRP) method that uses a non-monotone line search to ensure global convergence without explicit Jacobians. The RDF-PRP algorithm updates via $X_{k+1}=R_{X_k}(\alpha_k\Delta X_k)$ with search directions built from current and previous vector-field evaluations and vector transport, avoiding gradient/Jacobian calculations. A convergence theory shows that, under mild assumptions, either $\|F(X_k)\|\to 0$ or accumulation points satisfy $\langle JF(X_*)[F(X_*)],F(X_*)\rangle =0$, with stronger monotonicity implying convergence to zeros. To boost efficiency, a hybrid scheme combines RDF-PRP with a Riemannian Newton step, solving the Newton equation inexactly via CG to achieve high-precision zeros on problems including Oja's vector field, trace-ratio tangent fields, and monotone fields on Hadamard manifolds, demonstrating robustness and scalability to large-scale instances.
Abstract
This paper is concerned with the problem of finding a zero of a tangent vector field on a Riemannian manifold. We first reformulate the problem as an equivalent Riemannian optimization problem. Then we propose a Riemannian derivative-free Polak-Ribiére-Polyak method for solving the Riemannian optimization problem, where a non-monotone line search is employed. The global convergence of the proposed method is established under some mild assumptions. To further improve the efficiency, we also provide a hybrid method, which combines the proposed geometric method with the Riemannian Newton method. Finally, some numerical experiments are reported to illustrate the efficiency of the proposed method.