Regularization Algorithms for Solving Ill-Conditioned Matrix Problems in Inverse Problem Solving

In inverse problem solving, ill-conditioned matrix problems represent a common computational challenge. To address this issue, various regularization algorithms have been developed. The fundamental approach of these algorithms involves modifying the matrix or introducing specific constraints to ensure matrix stability and invertibility. Among the most widely used algorithms are Tikhonov regularization, Lasso regularization, and Ridge regularization. These methods not only resolve ill-conditioned matrix problems but are also extensively applied in fields such as statistics, machine learning, and deep learning. The Tikhonov regularization algorithm typically implements a smoothness constraint by adding a regularization term λI to the coefficient matrix, where λ represents the regularization parameter that controls the trade-off between solution stability and fitting accuracy. The mathematical implementation involves solving (AᵀA + λI)x = Aᵀb. Lasso regularization (L1 regularization) promotes sparsity in solutions by adding an L1-norm penalty term, making it particularly useful for feature selection problems. Its optimization objective function minimizes ||Ax - b||² + λ||x||₁, often solved using coordinate descent algorithms. Ridge regularization (L2 regularization) improves conditioning by adding an L2-norm penalty term to the diagonal elements of the matrix, implementing the solution through (AᵀA + λI)⁻¹Aᵀb. This approach effectively reduces solution variance while introducing some bias. These regularization techniques can be implemented using numerical computing libraries like NumPy or MATLAB, with key functions including matrix inversion operations, parameter optimization, and cross-validation for optimal regularization parameter selection.