We can think of Joan’s ratings as a weighted sum of these customers’ ratings. To do that, consider $m$ ($m \le 7200$) users whose ratings on the 100 jokes are known to us. For a new customer Joan who has rated 25 jokes, we want to be able to know how she would like the remaining 75 jokes and recommend her the one with the highest predicted rating. Suppose that we work for a company that makes joke recommendations to customers based on their known ratings. Each observation is a joke and each feature is a known rating on that joke from an existing user, at a scale of -10 to 10. The sample contains 100 observations and 7200 features and it is available here. To verify our findings, we will use a subsample of the Jester Datasets. Least squares can be described as follows: given the feature matrix $X$ of shape $n \times p$ and the target vector $y$ of shape $n \times 1$, we want to find a coefficient vector $\hat It is heavily based on Professor Rebecca Willet’s course Mathematical Foundations of Machine Learning and it assumes basic knowledge of linear algebra. This blog discusses the difference in least-squares weight vectors across over- and underdetermined linear systems, and how singular value decomposition (SVD) can be applied to derive a consistent expression.
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