There are many applications of the low-rank signal-plus-noise model 𝒀 = 𝑿 + 𝒁 where 𝑿 is a low-rank matrix and 𝒁 is noise, such as denoising and dimensionality reduction. We are interested in the properties of the latent matrix 𝑿, such as its singular value decomposition (SVD), but all we are given is the noisy matrix 𝒀. It is important to understand how the SVD components of 𝒀 relate to those of 𝑿 in the presence of a random noise matrix 𝒁. The field of random matrix theory (RMT) provides insights into those relationships, and this chapter summarizes some key results from RMT that help explain how the noise in 𝒁 perturbs the SVD components, by analyzing limits as matrix dimensions increase. The perturbations considered include roundoff error, additive Gaussian noise, outliers, and missing data. This is the only chapter that requires familiarity with the distributions of continuous random variables, and it provides many pointers to the literature on this modern topic, along with several demos that illustrate remarkable agreement between the asymptotic predictions and the empirical performance even for modest matrix sizes.

Chapter 2 starts out with a physics motivation, chosen to be the experimental study of the photoelectric effect by Millikan. It then discusses the concepts of absolute and relative error, as well as eror propagation in general. It then goes on to discuss the representation of real numbers on the computer, touching on topics like overflow, machine precision, and catastrophic cancellation. It then addresses a variety of problems where rounding error becomes significant: compensated summation, analytic manipulation, Taylor series, and recursion. The chapter is rounded out by a physics project, which studies the multipole expansion in electromagnetism, and a problem set. The physics project provides an opportunity to introduce Legendre polynomials, which play an important role in the following chapters.