MATLAB Implementation of GCV Function for Regularization Parameter Selection

The GCV (Generalized Cross-Validation) function serves as an effective tool in MATLAB for determining optimal regularization parameters, particularly when dealing with ill-posed problems or inverse problems. This function is typically integrated within the RegTools toolbox, automatically selecting the best regularization parameter by minimizing the generalized cross-validation function, thereby balancing solution accuracy with numerical stability. In regularization problems, appropriate parameter selection is crucial. The GCV method approximates leave-one-out cross-validation while avoiding the high computational complexity of direct residual calculations. The core algorithm evaluates prediction errors across different regularization parameters and selects the parameter that minimizes the GCV function value. This approach is particularly suitable for scenarios like Tikhonov regularization, where it effectively suppresses overfitting caused by noise. When implementing GCV in MATLAB, key considerations include proper data preprocessing and appropriate regularization matrix configuration. The function typically works alongside other regularization tools such as `l_curve`, providing automated parameter selection solutions for inverse problems. From a coding perspective, the implementation involves calculating the influence matrix and trace computations, with the GCV score typically computed using the formula: GCV(k) = (n*||Ax-b||^2) / (trace(I - A*A_k^†)^2), where A_k^† represents the regularized pseudo-inverse. The MATLAB implementation often utilizes built-in functions like `svd` for singular value decomposition and matrix operations for efficient computation, making it particularly valuable for large-scale inverse problems in signal processing and image reconstruction applications.