A Geometry-Adaptive Regularized Newton-Type Method for Manifold-Affine Intersection Problems

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, we prove local linear convergence to the intersection. Under transversality, we show that a residual-dependent choice of the regularization parameter yields higher-order local convergence; in particular, the method converges quadratically for the linear residual rule. We also analyze an inexact variant based on quasioptimal manifold projections. When the quasioptimality constant is sufficiently accurate, the inexact method retains local residual convergence. Numerical experiments on constructed degenerate SLRA instances and Hankel-structured examples illustrate the robustness of RN-SLRA in settings where Newton-SLRA may fail, and show that the inexact variant can reduce the projection cost in large-scale problems.