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.

This chapter discusses the important problem of matrix completion, where we know some, but not all, elements of a matrix and want to “complete” the matrix by filling in the missing entries. This problem is ill posed in general because one could assign arbitrary values to the missing entries, unless one assumes some model for the matrix elements. The most common model is that the matrix is low rank, an assumption that is reasonable in many applications. The chapter defines the problem and describes an alternating projection approach for noiseless data. It discusses algorithms for the practical case of missing and noisy data. It extends the methods to consider the effects of outliers with the robust principal component method, and applies this to video foreground/background separation. It describes nonnegative matrix factorization, including the case of missing data. A particularly famous application of low-rank matrix completion is the “Netflix problem”; this topic is also relevant to dynamic magnetic resonance image reconstruction, and numerous other applications with missing data (incomplete observations).