Alternating Least Squares Regression

Matrix decomposition

Consider a movie recommendation system with $n$ users and $m$ movies. Suppose the matrix $A \in \R^{n \times m}$ represents the rating matrix where its $(i,j)^{th}$ entry $A_{ij}$ represents the rating given by the $i^{th}$ user to the $j^{th}$ movie. We would like to represent/decompose the matrix $A$ as a product of the two matrices $X \in \R^{n \times r}, Y \in \R^{m \times r}$ as follows: