Centered Weak Discrete Riemannian Gradients: A Unified Framework for Riemannian Optimization

Abstract

We introduce the centered weak discrete Riemannian gradient (c-wDRG) framework for the unified analysis of optimization methods on Riemannian manifolds. The framework uses a center point to represent the relevant logarithmic differences in a common tangent space and covers Riemannian steepest descent, proximal point, proximal gradient, implicit midpoint, geodesic average-vector-field, Gonzalez, and Itoh--Abe methods. We derive c-wDRG certificates for these schemes and establish curvature-aware convergence results using explicit metric distortion bounds. The framework yields sublinear and linear convergence for nonaccelerated methods in the geodesically convex and strongly convex settings, respectively. We further develop accelerated c-wDRG schemes, obtaining accelerated sublinear convergence in the convex case and accelerated linear convergence in the strongly convex case. Our results provide explicit curvature-dependent stepsize conditions and convergence rates within a common framework for Riemannian first-order and discrete-gradient methods.