Moving Average-Based Variable Projection for Separable Nonlinear Problems

The identification of separable nonlinear models, prevalent in tasks such as signal analysis, image processing, time series analysis, and machine learning, presents a non-convex optimization challenge that necessitates the development of efficient identification algorithms. The Variable Projection (VP) algorithm has been proven to be quite effective for addressing these problems; however, traditional VP relying on the Hessian matrix and its inverse are highly time-consuming and unsuitable for complex, large-scale applications. This letter introduces a novel approach that employs the exponential moving average of gradient and gradient estimation bias to indirectly estimate the curvature of the objective landscape, proposing a Moving Average-based Variable Projection method (MAVP). The proposed algorithm utilizes only gradient information and can properly tackle the coupling relationships between different parameters during the optimization process, thereby achieving faster convergence. Numerical results on nonlinear time series analysis and image reconstruction demonstrate that the MAVP algorithm exhibits significant efficiency and effectiveness.