Regularization of Ill-Conditioned Equations

In this article, we discuss the regularization of ill-conditioned equations and present methods for implementing regularization algorithms using MATLAB. Regularization is a crucial technique that helps address ill-posed problems, enabling better handling of real-world data in practical applications. To better understand this concept, let's first briefly introduce ill-posed problems. In mathematics, an ill-posed problem refers to a situation where small changes in input data lead to significant variations in output results. This phenomenon is common in practical applications such as signal processing and image processing. To mitigate this issue, we can employ regularization techniques that constrain model complexity, thereby reducing the risk of ill-posed problems. In regularization algorithms, we can use various methods to limit model complexity, such as L1 regularization (Lasso) and L2 regularization (Ridge). These approaches help improve data handling in real-world applications while enhancing model accuracy. In MATLAB implementation, key functions like lasso() and ridge() can be utilized with appropriate parameter tuning to achieve optimal regularization effects. The algorithm typically involves adding a penalty term to the loss function, where L1 regularization promotes sparsity by adding the absolute value of coefficients, while L2 regularization adds the squared magnitude of coefficients to prevent overfitting.