{
  "abstract": "We propose RIAG-R, a Riemannian Inertial Adaptive Gradient method with gradient-triggered Restart, for optimization on Riemannian manifolds. RIAG-R combines three ingredients: an AdaGrad-type adaptive step-size that requires no knowledge of the Lipschitz constant; inertial extrapolation in the tangent space via the exponential map to accelerate convergence; and a restart rule, costing only one extra inner product per iteration, that detects and corrects the momentum-saturation pathology induced by manifold curvature. We establish a per-iteration descent inequality, an optimal $O(\\varepsilon^{-2})$-iteration complexity, a convergence result for cluster points of the generated sequence, and a linear convergence rate under the Riemannian Polyak-Łojasiewicz inequality. Numerical experiments on four representative matrix manifold optimization problems including sphere, Stiefel, symmetric positive-definite, and Grassmann manifolds demonstrate that RIAG-R consistently outperforms non-inertial and restart-free inertial baselines, confirming that adaptive restart is essential to realizing the benefits of momentum on curved spaces.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32997",
  "author_total": 3,
  "authors": [
    {
      "name": "Lateef O. Jolaoso"
    },
    {
      "name": "Professer V. Ndlovu"
    },
    {
      "name": "Maggie Aphane"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b530geq2jmkvb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32997",
  "fulltext_url": null,
  "id": "b530geq2jmkvb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32997",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2612|Advanced Optimization Algorithms Research"
    ]
  ],
  "openalex": "W7214722796",
  "pdf_url": "https://arxiv.org/pdf/2609.32997",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b530geq2jmkvb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32997",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32997"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "A Riemannian Inertial Adaptive Gradient Method with Momentum Restart",
  "title_text": "A Riemannian Inertial Adaptive Gradient Method with Momentum Restart",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/b530geq2jmkvb",
  "venue": "arXiv",
  "versions": []
}