Higher-Order Geometric Updates for Levenberg-Marquardt Method via Riemann Normal Coordinates
Abstract
Nonlinear least-squares objectives form the foundation of scientific machine-learning tasks, yet even curvature-aware optimizers remain geometrically inconsistent at finite step sizes: Levenberg-Marquardt (LM) derives its direction from local Riemannian metrics but realizes it as a straight parameter update. Here we introduce RNC-LM, which carries the LM direction along a locally constructed curved trajectory in Riemann normal coordinates. A recursive reformulation of the geodesic equation generates arbitrary finite-order corrections while reusing the same damped Gauss-Newton matrix factorization, and curve length is controlled separately from damping. On a reaction-diffusion physics-informed neural network benchmark, RNC-LM reduces relative L2 errors below 8×10−3, whereas L-BFGS, LM and LM with geodesic acceleration remain near one. On a large-scale machine-learning potential fitting task with 985,160 configurations, fourth-order RNC-LM reaches a fixed training-error target with a 34× wall-clock speedup over LM. These results establish finite-step geometric realization as a distinct optimizer design principle.
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BibTeX
@misc{liu2026higherorder,
title = {{Higher-Order Geometric Updates for Levenberg-Marquardt Method via Riemann Normal Coordinates}},
author = {Jianing Liu and Dong H. Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.07623},
eprint = {2607.07623},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.07623},
}
RIS
TY - UNPB
TI - Higher-Order Geometric Updates for Levenberg-Marquardt Method via Riemann Normal Coordinates
AU - Jianing Liu
AU - Dong H. Zhang
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2607.07623
UR - https://arxiv.org/abs/2607.07623
AB - Nonlinear least-squares objectives form the foundation of scientific machine-learning tasks, yet even curvature-aware optimizers remain geometrically inconsistent at finite step sizes: Levenberg-Marquardt (LM) derives its direction from local Riemannian metrics but realizes it as a straight parameter update. Here we introduce RNC-LM, which carries the LM direction along a locally constructed curved trajectory in Riemann normal coordinates. A recursive reformulation of the geodesic equation generates arbitrary finite-order corrections while reusing the same damped Gauss-Newton matrix factorization, and curve length is controlled separately from damping. On a reaction-diffusion physics-informed neural network benchmark, RNC-LM reduces relative L^2 errors below 8×10^−3, whereas L-BFGS, LM and LM with geodesic acceleration remain near one. On a large-scale machine-learning potential fitting task with 985,160 configurations, fourth-order RNC-LM reaches a fixed training-error target with a 34× wall-clock speedup over LM. These results establish finite-step geometric realization as a distinct optimizer design principle.
ER -