Optimization· 2026Q1
A Riemannian heavy-ball method with global convergence under the Riemannian Polyak–Łojasiewicz condition
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- 2026year
Short summary
A new Riemannian heavy-ball method guarantees global convergence for optimization problems on manifolds, even for non-convex objectives, by leveraging the weaker Polyak–Łojasiewicz condition instead of strong convexity.
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Key points
- Introduces a Riemannian heavy-ball method applicable to manifolds using retractions and vector transports.
- Establishes global convergence for general smooth, non-convex objectives: update directions are square summable and gradient norm vanishes along a subsequence.
- Proves global linear convergence rate in function values under the Riemannian Polyak–Łojasiewicz (PL) inequality, a condition weaker than strong convexity.
- Demonstrates practical performance via numerical experiments on sparse PCA and sphere-constrained signal denoising.
AI-generated from the title and abstract; the full text is not read.
Abstract
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.
The authors' abstract, as published at the source. Optimization, 2026 · DOI ↗
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Field: Numerical Analysis
Numerical AnalysisMathematics