An Efficient Algorithm for Nonlinear Regression Problems With Huber Loss

Guangyong Chen

doi:10.1109/tac.2026.3683294

An Efficient Algorithm for Nonlinear Regression Problems With Huber Loss

2026-04-13

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陈晨 (C. L. Philip Chen)

DOI：10.1109/tac.2026.3683294

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Nonlinear regression models are essential in various fields, such as system identification, data analysis, and signal processing. Despite their importance, the efficiency of robust parameter estimation is hindered by inherent nonlinearity and nonsmoothness. This article presents an efficient and robust parameter estimation algorithm for nonlinear regression problems by utilizing the smooth Huber loss instead of the $L_{1}$ loss. We begin by analyzing the structural characteristics of nonlinear regression with Huber loss, partitioning the parameters into linear and nonlinear components. A key theorem regarding residuals is then established, which aids in addressing the coupling between parameters during optimization. Building on these insights, we introduce the Huber variable projection (HuVP) algorithm, a novel variable projection method that efficiently handles parameter coupling and optimizes in lower dimensional spaces. The convergence properties of HuVP are analyzed, showing that it inherits the advantages of the traditional variable projection algorithm. Numerical experiments demonstrate the effectiveness and robustness of HuVP across different scenarios.

Keywords:

Nonlinear regression

robust parameter estimation

system identification

variable projection (VP) algorithm

An Efficient Algorithm for Nonlinear Regression Problems With Huber Loss

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