Polynomial Acceleration of Alternating Projections: Chebyshev Acceleration, Convergence Rates, and Retractions

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∈(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↓0, versus O(θ_F⁻²) for ordinary AP.}} {The framework also covers} inexact, relaxed, and generalized AP {under their respective regularity and stability conditions.}

Acceleration of the Method of Alternating Projections on subspaces by GMRES
Luis Manuel Hernández-Ramos, Maryam Calles-Cabriles, Gorka Emaldi-Porco
arXiv27 Oct 2026PreprintarXiv licenceFull textNumerical analysis & scientific computingarXiv
A new convergence proof for an iterative scheme for elliptic Cauchy problems based on alternating orthogonal projections
Abhijit Guchhait, Adrian Muntean, Martin Lind
arXiv10 Oct 2026PreprintCC BY 4.0Full textAnalysis & PDEs Numerical analysis & scientific computingarXiv
Deeply unrolled alternating projections for phase retrieval
Jiayi Yao, Vasileios Charisopoulos, Rebecca Willett
arXiv12 Oct 2026PreprintarXiv licenceFull textStatisticsarXiv
Volume Sampling and Spectral Equalization in Randomized Alternating Projections
Alireza Entezari, Arunava Banerjee, Leila Kalantari
arXiv8 Oct 2026PreprintCC BY-NC-ND 4.0Full textNumerical analysis & scientific computingarXiv
The Links--Gould polynomial of an alternating link is alternating
Matthew Harper, Ben-Michael Kohli
arXiv13 Oct 2026PreprintCC BY 4.0Full textGeometry & topologyarXiv

Cite

BibTeX RIS

BibTeX

@misc{zheng2026polynomial,
  title = {{Polynomial Acceleration of Alternating Projections: Chebyshev Acceleration, Convergence Rates, and Retractions}},
  author = {Dengyu Zheng and Shixiang Chen},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.06150},
  eprint = {2610.06150},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.06150},
}

RIS

TY  - UNPB
TI  - Polynomial Acceleration of Alternating Projections: Chebyshev Acceleration, Convergence Rates, and Retractions
AU  - Dengyu Zheng
AU  - Shixiang Chen
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.06150
UR  - https://arxiv.org/abs/2610.06150
AB  - 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∈(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↓0, versus O(θ_F^−2) for ordinary AP.}} {The framework also covers} inexact, relaxed, and generalized AP {under their respective regularity and stability conditions.}
ER  -

Abstract

Related

Cite