Lecture 4: Designing linear systems; special structure (part I)

The instructor addresses student questions regarding homework assignments, clarifies minor errors, and encourages students to utilize office hours for assistance, highlighting the importance of proactive engagement with the course material. This lecture on numerical analysis covers linear systems of equations, focusing on least squares solutions for overdetermined systems (more equations than unknowns). It introduces regression, polynomial fitting, and Fourier transforms as applications. The lecture emphasizes the normal equations (AᵀAx = Aᵀb) for least squares, discusses overfitting and regularization techniques (Tikhonov/ridge regression, lasso, elastic net) to address ill-conditioned matrices and noisy data, improving solution stability. The instructor announces a change in lecture structure, prioritizing applications and motivations before delving into detailed numerical methods. This segment emphasizes the practical relevance of the course material to computer science.The instructor introduces the concept of parametric regression, explaining its goal of approximating functions and differentiating it from non-parametric regression. The explanation sets the stage for subsequent examples and problem-solving. This segment expands on the concept of regression by demonstrating how the same methodology can be applied to polynomial and trigonometric functions, showcasing the versatility of the approach and connecting it to familiar concepts like Fourier transforms. The instructor discusses overdetermined linear systems, where the number of equations exceeds the number of unknowns, introducing the concept of overfitting and the need for approximation. The segment builds anticipation for the derivation of the normal equations for least squares. This segment details the derivation of normal equations (AᵀAx = Aᵀb) for solving least squares problems, highlighting their importance in linear algebra and emphasizing common mistakes in their application, particularly concerning the invertibility of AᵀA and the distinction between necessary and sufficient conditions for minimization. The discussion includes examples and warnings against common misconceptions. This segment introduces regularization techniques, specifically focusing on Tikhonov regularization (ridge regression), Lasso, and elastic net, as solutions to handle underdetermined systems or noisy data in least squares problems. It explains how these methods address the issue of non-unique solutions by adding penalty terms to the objective function, impacting the solution's properties and sparsity. This segment delves into the practical implementation of Tikhonov regularization, showing how it modifies the normal equations by adding a term to the diagonal of the matrix. It explains the impact on numerical stability and condition number, improving the solution's robustness, especially when dealing with ill-conditioned systems. The discussion also touches upon the Bayesian interpretation of regularization and its effect on the solution's properties.