A Riemannian heavy-ball method with global convergence under the Riemannian Polyak–Łojasiewicz condition

The heavy-ball (HB) method is a classical momentum scheme for large-scale optimization, with well-known fast local behaviour under smoothness and strong convexity in Euclidean spaces. We develop a Riemannian heavy-ball method on manifolds equipped with a retraction and an associated vector transport, allowing practical implementations beyond geodesic updates. Our analysis does not rely on strong convexity, for general smooth (possibly nonconvex) objectives, we establish global convergence in the sense that the update directions are square summable and the Riemannian gradient norm vanishes along a subsequence. Under the Riemannian Polyak–Łojasiewicz (PL) inequality, a gradient-dominance condition strictly weaker than strong convexity, we further prove a global linear convergence rate in function values. These guarantees also specialize to the Euclidean setting and yield nonconvex global results for HB under the PL condition. Numerical experiments implemented with Manopt illustrate the practical performance of the method on manifold-structured tasks, including sparse PCA via Stiefel-type formulations and sphere-constrained signal denoising.