This chapter covers principal component analysis and low-rank models, which are popular techniques to process high-dimensional datasets with many features. We begin by defining the mean of random vectors and random matrices. Then, we introduce the covariance matrix which encodes the variance of any linear combination of the entries in a random vector, and explain how to estimate it from data. We model the geographic location of Canadian cities as a running example. Next, we present principal component analysis (PCA), a method to extract the directions of maximum variance in a dataset. We explain how to use PCA to find optimal low-dimensional representations of high-dimensional data and apply it to a dataset of human faces. Then, we introduce low-rank models for matrix-valued data and describe how to fit them using the singular-value decomposition. We show that this approach is able to automatically identify meaningful patterns in real-world weather data. Finally, we explain how to estimate missing entries in a matrix under a low-rank assumption and apply this methodology to predict movie ratings via collaborative filtering.

Analysis of various data sets can be accomplished using techniques based on least-squares methods.For example, linear regression of data determines the best-fit line to the data via a least-squares approach.The same is true for polynomial and regression methods using other basis functions.Curve fitting is used to determine the best-fit line or curve to a particular set of data, while interpolation is used to determine a curve that passes through all of the data points.Polynomial and spline interpolation are discussed.State estimation is covered using techniques based on least-squares methods.

In this chapter we shall explore the idea of dimensionality reduction in more detail. We begin with a discussion of eigenvalues and their use in “principal component analysis” (PCA). We cover singular-value decomposition, a more powerful version of UV-decomposition. Finally, because we are always interested in the largest data sizes we can handle, we look at another form of decomposition, called CUR-decomposition, which is a variant of singular-value decomposition that keeps the matrices of the decomposition sparse if the original matrix is sparse.