maximum-Margin Matrix Factorization

Assume Y as a n by m matrix containing n user preferences about m movies, such that [math]\displaystyle{ Y_{ij} = +1 }[/math] if user i likes movie j, and [math]\displaystyle{ Y_{ij} = -1 }[/math] if he dislikes it. Due to lack of knowledge about the users’ opinions, Y is partially observable: it Y has some [math]\displaystyle{ pm }[/math] and the other cells are unknown. The main goal is to find matrix X such than it preserves the knowledge in Y, and predicts the value of its unknown cells.