Probabilistic Matrix Factorization Task

Context:
- It can be solved by a Probabilistic Matrix Factorization System (that implements a Probabilistic Matrix Factorization Algorithm).
Example(s):
Counter-Example(s):
- an Exact Matrix Factorization Task.
See: Covariance Matrix Approximation.

References

http://wikicoursenote.com/wiki/Probabilistic_Matrix_Factorization
- QUOTE: Unlike all of the above-mentioned approaches with the exception of the matrix-factorization-based ones, PMF scales well to large datasets. Furthermore, unlike most of the existing algorithms, which have trouble making accurate predictions for users who have very few ratings, PMF performs well on very sparse and imbalanced datasets, such as the Netflix dataset.
  Given the preference matrix described above with entries [math]\displaystyle{ \,R_{ij} }[/math], the aim is to find a factorization that minimizes the root mean squared error(RMSE) on the test set. An initial attempt is to use a linear model where we assume that there is Gaussian noise in the data. Define [math]\displaystyle{ \,I_{ij} }[/math] to be 1 if [math]\displaystyle{ \,R_{ij} }[/math] is known (i.e. user [math]\displaystyle{ \,i }[/math] has rated movie [math]\displaystyle{ \,j }[/math]) and 0 otherwise. Further, let [math]\displaystyle{ \,\mathcal N(x|\mu,\sigma^2) = f_X(x) }[/math] with [math]\displaystyle{ X \sim \mathcal N(\mu,\sigma^2) }[/math]. Then, we can define a conditional probability of the ratings with hyperparameter [math]\displaystyle{ \,\sigma^2 }[/math]

(Mnih & Salakhutdinov, 2007) ⇒ Andriy Mnih, and Ruslan Salakhutdinov. (2007). “Probabilistic Matrix Factorization.” In: Advances in Neural Information Processing Systems.
- QUOTE: Many existing approaches to collaborative filtering can neither handle very large datasets nor easily deal with users who have very few ratings. In this paper we present the Probabilistic Matrix Factorization (PMF) model which scales linearly with the number of observations and, more importantly, performs well on the large, sparse, and very imbalanced Netflix dataset.