The Intrinsic Riemannian Proximal Gradient Method for Convex Optimization

2025/07/21 by Ronny Bergmann, Hajg Jasa, Bergmann, Ronny +7 · 1 citation
Mathematics · Engineering · Computer Science · #Advanced Optimization Algorithms Research #Sparse and Compressive Sensing Techniques #Optimization and Variational Analysis

paper · pdf · doi:10.48550/arxiv.2507.16055

Abstract

We consider a class of (possibly strongly) geodesically convex optimization problems on Hadamard manifolds, where the objective function splits into the sum of a smooth and a possibly nonsmooth function. We introduce an intrinsic convex Riemannian proximal gradient (CRPG) method that employs the manifold proximal map for the nonsmooth step, without operating in the embedding or tangent space. A sublinear convergence rate for convex problems and a linear convergence rate for strongly convex problems is established, and we derive fundamental proximal gradient inequalities that generalize the Euclidean case. Our numerical experiments on hyperbolic spaces and manifolds of symmetric positive definite matrices demonstrate substantial computational advantages over existing methods.

Citations

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Cited by

• The Intrinsic Riemannian Proximal Gradient Method for Nonconvex Optimization
• A Riemannian Accelerated Proximal Gradient Method

Related

• The Intrinsic Riemannian Proximal Gradient Method for Nonconvex Optimization
• A Riemannian Accelerated Proximal Gradient Method
• An Adaptive Proximal Point Method for Nonsmooth and Nonconvex Optimization on Hadamard Manifolds
• Convergence and Trade-Offs in Riemannian Gradient Descent and Riemannian Proximal Point
• Riemannian Proximal Gradient Methods (extended version)