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.

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BibTeX RIS

BibTeX

@misc{zhou2026centered,
  title = {{Centered Weak Discrete Riemannian Gradients: A Unified Framework for Riemannian Optimization}},
  author = {Derun Zhou and Mahito Sugiyama},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.21862},
  eprint = {2608.21862},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.21862},
}

RIS

TY  - UNPB
TI  - Centered Weak Discrete Riemannian Gradients: A Unified Framework for Riemannian Optimization
AU  - Derun Zhou
AU  - Mahito Sugiyama
PY  - 2026
DA  - 2026/10/03
PB  - arXiv
DO  - 10.48550/arxiv.2608.21862
UR  - https://arxiv.org/abs/2608.21862
AB  - 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.
ER  -

Abstract

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