SVD-Preconditioned Gradient Descent Method for Solving Nonlinear Least Squares Problems

• URL: http://arxiv.org/abs/2602.09057v1
• Date: Sat, 07 Feb 2026 18:53:00 GMT
• Title: SVD-Preconditioned Gradient Descent Method for Solving Nonlinear Least Squares Problems
• Authors: Zhipeng Chang, Wenrui Hao, Nian Liu,
• Abstract summary: This paper introduces a novel optimization algorithm designed for nonlinear least-squares problems.<n>The method is derived by preconditioning the gradient descent direction using the Singular Value Decomposition (SVD) of the Jacobian.<n>We establish the local linear convergence of the proposed method under standard regularity assumptions and prove global convergence for a modified version of the algorithm under suitable conditions.
• Score: 27.21342746802453
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This paper introduces a novel optimization algorithm designed for nonlinear least-squares problems. The method is derived by preconditioning the gradient descent direction using the Singular Value Decomposition (SVD) of the Jacobian. This SVD-based preconditioner is then integrated with the first- and second-moment adaptive learning rate mechanism of the Adam optimizer. We establish the local linear convergence of the proposed method under standard regularity assumptions and prove global convergence for a modified version of the algorithm under suitable conditions. The effectiveness of the approach is demonstrated experimentally across a range of tasks, including function approximation, partial differential equation (PDE) solving, and image classification on the CIFAR-10 dataset. Results show that the proposed method consistently outperforms standard Adam, achieving faster convergence and lower error in both regression and classification settings.

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