RECENT APPLICATIONS OF PCA AND SVD

Abstract

In this thesis, Principal Component Analysis is a powerful dimensionality reduction technique that transforms high-dimensional data into a lower-dimensional space while preserving variance. By comput- ing the covariance matrix and its eigenvectors, PCA finds principal components that best represent the data. It is used on large scale in image compression, face recognition, and feature extraction, simplifying complex datasets without losing critical information. SVD is a matrix factorization method that decom- poses any matrix into three distinct matrices: A = U ΣV T . This decomposition reveals hidden patterns in data and has applications in data compression, noise reduction, and recommendation systems. Unlike PCA, which relies on eigenvectors of the covariance matrix, SVD works directly on the data matrix, making it more versatile. PCA and SVD are two fundamental techniques in linear algebra that have revolutionized data science, machine learning, and image processing. This presentation explores their mathematical foundations, geometric interpretations, and real-world applications.