A Regularized Newton-Type Method for Manifold--Affine Intersection Problems under Intrinsic Transversality
Abstract
We propose Regularized Newton-SLRA (RN-SLRA), a regularized Newton-type method for local manifold--affine intersection problems motivated by structured low-rank approximation. Classical Newton-SLRA achieves fast local convergence under transversality, but its tangent-space intersection step may become ill-defined, singular, or severely ill-conditioned when transversality fails. RN-SLRA overcomes this difficulty by replacing the exact tangent-space intersection step with a regularized quadratic subproblem over the affine space. Under intrinsic transversality, RN-SLRA with regularization parameter μk=c rkρ converges locally with Q-order at least 1+ρ for 0<ρ≤1, including quadratically for ρ=1, and linearly for ρ=0. We also analyze an inexact variant based on σ-quasioptimal manifold projections. It retains the 1+ρ order for 0<ρ≤1, and converges linearly for fixed regularization when qσ<1, where q is the local one-step alternating-projection factor. Experiments support the predicted quadratic behavior on an analytically clean but nontransversal SLRA instance, where the classical Newton tangent system is severely ill-conditioned, and show reduced projection costs for the inexact method on large-scale Hankel problems.
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BibTeX
@misc{zheng2026regularized,
title = {{A Regularized Newton-Type Method for Manifold--Affine Intersection Problems under Intrinsic Transversality}},
author = {Dengyu Zheng and Shixiang Chen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.31738},
eprint = {2606.31738},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.31738},
}
RIS
TY - UNPB
TI - A Regularized Newton-Type Method for Manifold--Affine Intersection Problems under Intrinsic Transversality
AU - Dengyu Zheng
AU - Shixiang Chen
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2606.31738
UR - https://arxiv.org/abs/2606.31738
AB - We propose Regularized Newton-SLRA (RN-SLRA), a regularized Newton-type method for local manifold--affine intersection problems motivated by structured low-rank approximation. Classical Newton-SLRA achieves fast local convergence under transversality, but its tangent-space intersection step may become ill-defined, singular, or severely ill-conditioned when transversality fails. RN-SLRA overcomes this difficulty by replacing the exact tangent-space intersection step with a regularized quadratic subproblem over the affine space. Under intrinsic transversality, RN-SLRA with regularization parameter μ_k=c r_k^ρ converges locally with Q-order at least 1+ρ for 0<ρ≤1, including quadratically for ρ=1, and linearly for ρ=0. We also analyze an inexact variant based on σ-quasioptimal manifold projections. It retains the 1+ρ order for 0<ρ≤1, and converges linearly for fixed regularization when qσ<1, where q is the local one-step alternating-projection factor. Experiments support the predicted quadratic behavior on an analytically clean but nontransversal SLRA instance, where the classical Newton tangent system is severely ill-conditioned, and show reduced projection costs for the inexact method on large-scale Hankel problems.
ER -