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 convergence bounds with explicit curvature-dependent step-size conditions. Under suitable c-wDRG parameter and step-size conditions, nonaccelerated schemes achieve \(O(k^{-1})\) or linear convergence of the objective gap. We further develop accelerated c-wDRG schemes with \(O(k^{-2})\) or accelerated linear convergence under the respective parameter conditions. Numerical experiments support the theoretical convergence results.