{
  "abstract": "Alternating projections (AP) near a clean intersection of smooth manifolds can converge slowly when the Friedrichs angle is small. We develop the \\emph{Fixed-Point Manifold Acceleration (FPMA) framework} for {local convergence and polynomial acceleration of} smooth maps that fix a manifold pointwise. {Under suitable regularity and uniform normal spectral stability, FPMA establishes local R-linear convergence and a smooth limiting map inducing a local retraction. At fixed points, consistent polynomial acceleration with $p(1)=1$ preserves the selected limit's first-order response to normal initial perturbations whenever the transformed normal blocks are uniformly spectrally stable. Under sufficient smoothness, we give a curvature criterion for second-order induced retractions and show that stable consistent polynomial acceleration preserves both the quadratic retraction expansion and this criterion. {As a special case, we establish new local R-linear convergence guarantees for fixed-degree Chebyshev-accelerated exact AP and second-order induced retractions. For uniform Friedrichs angle $θ_F\\in(0,π/2)$, acceleration strictly improves the local convergence rate per AP evaluation. The large-degree limiting spectral factor at fixed $θ_F$ predicts $O(θ_F^{-1})$ evaluations for fixed relative error reduction as $θ_F\\downarrow0$, versus $O(θ_F^{-2})$ for ordinary AP.}} {The framework also covers} inexact, relaxed, and generalized AP {under their respective regularity and stability conditions.}",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06150",
  "author_total": 2,
  "authors": [
    {
      "name": "Dengyu Zheng"
    },
    {
      "name": "Shixiang Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/90ydfk2qtk9dz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06150",
  "fulltext_url": null,
  "id": "90ydfk2qtk9dz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06150",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1703|Optimization and Variational Analysis"
    ]
  ],
  "openalex": "W7220728319",
  "pdf_url": "https://arxiv.org/pdf/2610.06150",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/90ydfk2qtk9dz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.06150",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06150"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Polynomial Acceleration of Alternating Projections: Chebyshev Acceleration, Convergence Rates, and Retractions",
  "title_text": "Polynomial Acceleration of Alternating Projections: Chebyshev Acceleration, Convergence Rates, and Retractions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/90ydfk2qtk9dz",
  "venue": "arXiv",
  "versions": []
}