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    This study introduces an efficient and robust variable projection (VP) algorithm for separable nonlinear models. The new method uses analytical Jacobian matrices and the modified Gram-Schmidt (MGS) approach for improved performance in machine learning and system identification.

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    Area of Science:

    • Numerical analysis
    • Optimization algorithms
    • Machine learning

    Background:

    • Separable nonlinear models are widely used in machine learning and system identification.
    • The variable projection (VP) approach offers an efficient method for optimizing these models.
    • Existing VP algorithms often rely on finite differences for Jacobian matrix approximation.

    Purpose of the Study:

    • To investigate various variable projection (VP) algorithms utilizing different matrix decompositions.
    • To enhance the efficiency of VP algorithms by employing analytical Jacobian matrices instead of finite differences.
    • To introduce a more robust VP algorithm implementation for separable nonlinear least-squares problems using the modified Gram-Schmidt (MGS) method.

    Main Methods:

    • Exploration of VP algorithms based on diverse matrix decomposition techniques.
    • Implementation of analytical Jacobian matrix calculations to replace finite difference approximations.
    • Development of a VP algorithm variant incorporating the modified Gram-Schmidt (MGS) method.

    Main Results:

    • The use of analytical Jacobian matrices significantly improves VP algorithm efficiency.
    • The MGS-based VP algorithm demonstrates enhanced robustness for separable nonlinear least-squares problems.
    • Comparative numerical experiments validate the superior performance of the proposed MGS method-based VP algorithm over four other implementations.

    Conclusions:

    • The proposed MGS method-based VP algorithm offers a more efficient and robust solution for optimizing separable nonlinear models.
    • Employing analytical Jacobian matrices is a key factor in improving the computational efficiency of VP algorithms.
    • This research contributes a valuable tool for applications in machine learning and system identification.