Extending Self-Organizing Maps with uncertainty information of probabilistic PCA

We introduce a probabilistic version of the self-organizing map (SOM) where we model the uncertainty of both the model vectors and the data. While uncertainty information about the data is often not available, this property becomes very useful when the method is combined in a hierarchical manner with probabilistic principal component analysis (PCA), where we do estimate uncertainty of the principal components and the weights. We apply the hierarchical model to the domain of collaborative filtering, where probabilistic PCA is a popular approach due to its robustness for tackling many missing values in the data. The main focus in this paper is for recommendation systems about movies, where the movie rating data matrix of size people times movies is available, but contains lots of missing values. The matrix is first decomposed into a matrix product of people times features and features times movies by PCA. Then we apply the probabilistic SOM to both of those matrices separately. The uncertainty is large when a person (or a movie) has only a few ratings. The experiments with Movielens and Netflix data show an improvement over traditional SOM.