A robust nonlinear fitting framework with an adaptive smoothed L f loss and rank selection via singular value decomposition

A unified framework for robust nonlinear least-squares fitting is presented. The objective is an adaptive smoothed L f loss with exponent f > 0 and smoothing parameter ε ≥ 0. For f ≥ 1 it matches the f -th power of the ℓf norm up to a constant; for f < 1 it is a non-convex quasi-norm-type robust loss, and no global-optimality guarantee is claimed. The loss interpolates continuously between ordinary least squares ( f = 2), smoothed least absolute deviations ( f = 1), and increasingly aggressive robust losses ( f < 1). Minimization uses iteratively reweighted least squares (IRLS) inside a Levenberg–Marquardt loop with gain-ratio adaptive damping. Each step is taken on a low-rank subspace of the singular value decomposition (SVD) of the weighted Jacobian, chosen by a dual criterion combining energy-based truncation with gradient capture. The IRLS gradient is exact, and a majorization–minimization argument shows that every accepted step strictly decreases the true objective at fixed ( f , ε). Further mechanisms—a convex-to-robust warm-start schedule annealing f and ε, dual-direction sign detection (a heuristic), dual-strategy polishing, and a residual-adaptive target-exponent rule—are assessed empirically rather than proven. The evaluation covers all 27 nonlinear regression benchmarks of the National Institute of Standards and Technology Statistical Reference Datasets (NIST StRD) under three contamination rates with 20 random seeds per scenario, plus four nonlinear synthetic models with three contamination types (Gaussian, uniform, one-sided), three outlier magnitudes, leverage points, and scaling to p = 50. Against seven baselines, each supplied with scale parameters based on the median absolute deviation (MAD) from a two-phase robust procedure, the solver attains the lowest mean relative error (RE) (0.334 vs . 37.2 next-best); as this aggregate is dominated by baseline catastrophic failures, geometric means, medians, interquartile ranges (IQR), worst cases, and paired significance tests are also reported. On typical trials it is statistically indistinguishable from the strongest robust baselines and slightly behind by median error (0.054 vs . 0.031–0.047 for Barron, Geman–McClure (GM), and Cauchy), but separates decisively in reliability: a 4.1% catastrophic failure rate vs . 11.0–37.5%, the smallest IQR (0.133 vs . 0.29–3.87), a bounded worst case (RE = 77 vs . 10 4 –10 12 ), and zero failures over 240 contaminated synthetic trials. The method is not uniformly superior: ordinary least squares is more accurate on several clean datasets, and Barron’s adaptive loss and the Cauchy estimator achieve lower error on some easier contaminated problems. Accuracy is stable across the contamination types and outlier magnitudes tested; invariance beyond these settings is not claimed.