High Dimensional Data

High-dimensional datasets are increasingly common in modern data analysis, especially in fields like genomics, image processing, natural language processing, and recommender systems. There is a variety of computational techniques and statistical concepts that are useful for analyzing datasets in which each observation is associated with a large number of numerical variables. In this part of the book, we introduce ideas that are useful in the analysis of these high-dimensional datasets. Specifically, we provide brief introductions to linear algebra, dimension reduction, matrix factorization, and regularization. As motivating examples, we use handwritten digit recognition and movie recommendation systems, both of which involve high-dimensional datasets with hundreds or thousands of variables per observation. We start this part of the book by demonstrating how to work with matrices in R.

One specific task we use to motivate linear algebra is measuring the similarity between two handwritten digits. Because each digit is represented by \(28 \times 28 = 784\) pixel values, we cannot simply subtract two vectors as we would in a one-dimensional setting. Instead, we treat each observation as a point in a high-dimensional space and use a mathematical definition of distance to quantify similarity. Many machine learning techniques introduced later in the book rely on this geometric interpretation.

We also use this high-dimensional concept of distance to motivate dimension reduction, a set of techniques that summarize high-dimensional data in lower-dimensional representations that are easier to visualize and analyze, while preserving the essential information. Distance between observations provides a concrete example: we aim to reduce the number of variables while preserving the pairwise distances between observations as much as possible. This leads naturally to matrix factorization methods, which arise from the mathematical structure underlying these techniques.

Finally, we introduce the concept of regularization, which is useful when analyzing high-dimensional data. In many applications, the large number of variables increases the risk of overfitting or cherry-picking results that appear significant by chance. Regularization provides a mathematically principled way to constrain models, improve generalization, and avoid misleading conclusions.

Together, these topics lay the groundwork for understanding and implementing many of the machine learning techniques we cover in the next part of the book.