Principal Component Analysis by Max Litvak '25

Principal Component Analysis by Max Litvak ’25, Wednesday November 6, 1:00 – 1:50pm, North Science Building 113, Wachenheim, Mathematics Colloquium

Abstract: PCA is a dimensionality reduction technique that has may applications. It is used for image compression, pre-processing data for machine learning, and any many other data analysis problems. The data is constructed as an n x p matrix, X, where n is the number of samples and p is the number of features. The goal of PCA is to find a p x y matrix of weights which converts X to an n x y matrix where y < p. The problem reduces to maximizing the Rayleigh Quotient of X(X)^T. The spectral Theorem, which I will prove, implies that the maximum Rayleigh Quotient is equivalent to the largest eigenvalue of X(X)^T. In this way, we can find the matrix of weights to transform our original matrix, X, into one with less dimensions.