Recall that each user a has a set of movies that $($s$)$he has already rated. Let $Y$ be a matrix with $n$ row and $m$ columns whose $(a,i{)}^{th}$ entry $Y_{ai}$ is the rating by user $a$ of movie $i$ if this rating has already been given, and blank if not. Our goal is to come up with a matrix $x$ that has no blank entries and whose $(a,i{)}^{th}$ entry $x_{ai}$ is the prediction of the rating user a will give to movie $i.$

Let $D$ be the set of all $(a,i)\text{'}$s for which a user rating $Y_{ai}$ exists, i$.$e$.(a,i)inD$ if and only if the rating of user $a$ to movie i exists.

A naive approach to solve this problem would be to minimize the following objective:

$J(x)=({\displaystyle \underset{a}{\overset{?}{}}},$iinD$\frac{(Y_{ai}-x_{ai}{)}^2}{2}+\frac{}{2}{\displaystyle \underset{(a)}{\overset{?}{}}},i{x}_{ai}^2$

Where the first term is the sum of the squared errors for entries with observed rating, and the second term is a regularization term roughly to prevent the predictions to become extremely large, and the parameter $$ controls the balance between theses two terms.

Compute the derivative $\frac{\text{d}\text{e}\text{l}\text{J}}{del{x}_{ai}}$ of the objective function $J(x).($Note that $J(x)$ can be viewed as a function of the variables $x_{ai.})$

$($Type X$\_\{$ai$\}$ for matrix entries $x_{ai},Y_{-}\{ai\}$ for matrix entries $Y_{ai}$ and "lambda" for $.$ Note that $x$ and $Y$ are