Lecture 9: Nonlinear equations and convergence analysis (part I)

The professor announces the upcoming midterm, covering numerical linear algebra and briefly reviews the singular value decomposition (SVD), highlighting its importance as a general factorization method applicable to various matrix types. The segment emphasizes the significance of understanding SVD for solving various computational problems. This segment introduces the problem of 3D point cloud registration using the Kinect as an example. The professor explains the challenge of aligning multiple point clouds obtained from different viewpoints and introduces the Procrustes problem as a method for solving this, highlighting its practical applications in computer vision and graphics. CS 205A midterm is next week, covering numerical linear algebra (error estimates, floating point math, nonlinear systems). The SVD is a powerful tool for various applications (region alignment, Procrustes problem, data analysis). Review Gaussian elimination, LU, Cholesky, QR factorizations, eigenvalue problems, and the SVD. Focus on recognizing problem types and applying appropriate methods. Optional homework offers extra late days. This segment showcases an application of SVD in computer graphics, specifically in mesh deformation. The professor explains how SVD is used to locally approximate deformations with orthogonal matrices, enabling users to manipulate 3D models in a realistic way. A YouTube video example is referenced to illustrate the concept.The professor connects SVD to a statistical problem of dimensionality reduction. The segment explains how SVD can be used to find the best low-dimensional subspace to project data points onto, minimizing information loss. The professor encourages students to review the proof in the lecture notes for a deeper understanding. This segment emphasizes the crucial skill of recognizing and reducing real-world problems (from vision to machine learning) into solvable numerical problems like eigenvalue problems or least squares. It stresses the importance of understanding the underlying principles rather than solely focusing on coding implementations, encouraging students to experiment with different methods and seek help when needed. This segment explains two strategies for QR factorization: Gram-Schmidt (easy but numerically unstable) and Householder (numerically principled). It highlights the advantage of QR factorization in solving least squares problems without squaring the condition number, contrasting it with the potential issues of LU factorization of normal equations.This segment delves into eigenvalue problems, introducing matrix factorization (A = XDX⁻¹) and the spectral theorem, emphasizing its importance for symmetric matrices. It discusses the power method, an iterative algorithm for computing eigenvectors, and its limitations when dealing with matrices lacking a full eigenspace. The segment concludes by mentioning the singular value decomposition (SVD) as a powerful tool built upon the foundation of eigenvectors.