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 ${\mu }_k=c{r}_k^{\rho }$ converges locally with Q-order at least $1+\rho$ for $0<\rho \le 1$, including quadratically for $\rho =1$, and linearly for $\rho =0$. We also analyze an inexact variant based on $\sigma$-quasioptimal manifold projections. It retains the $1+\rho$ order for $0<\rho \le 1$, and converges linearly for fixed regularization when $q\sigma <1$, 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.