In a real-world recommender system, the rating matrix will eventually become very large as more users are added to the system and the list of items being offered grows. As a result, we may want to apply a dimensionality reduction technique to this matrix. Ideally, we would like to retain as much information as possible from the original matrix while doing this. One such method that has applications across a wide range of disciplines uses singular value decomposition, or SVD as it is commonly abbreviated to.

SVD is a matrix factorization technique that has a number of useful applications, one of which is dimensionality reduction. It is related to the PCA method of dimensionality reduction that we saw in Chapter 1, Gearing Up for Predictive Modeling, and many people confuse the two. SVD actually describes just a mathematical method of factorizing matrices. In fact, some implementations of PCA use SVD to compute the principal components.

Let's begin by looking at...