In the era of big data, we are surrounded by recommendation systems that leverage and predict our responses, from daily shopping patterns to political campaigns and even presidential elections. There are various techniques and algorithms available for collaborative recommendation (in a broad sense, collaborative filtering), including memory based, regression based, objective function based, and so on. These systems are able to automatically suggest closely related options from vastly available information, beyond any human ability to manually deal with it. Nowadays, state-of-the-art recommendation systems provide fairly accurate recommendations in many areas. However, as the authors mention, optimizations are needed for recommender systems with, for example, multiple constraints. This paper proposes an optimization model with a covariance-based regularizer.

For a recommendation system with users u and items v, the proposed model starts with a sample matrix M_v×u, where m_ij represents the preference of user j on item i. The Mahalanobis distance between x_j, the j-th column of X, and M is defined as , where is the mean computed over the columns of M, and ∑^-1 is the inverse covariance matrix associated with M. The objective of a user-based system is to calculate the x_j that minimizes the Mahalanobis distance in the sense of least squares. Working with the transpose of M, one obtains an item-based system. To improve computational efficiency, the covariance-based regularizer applies two techniques on dimensionality reduction: matrix factorization and principal component analysis (PCA).

The experiments were conducted on four datasets: FilmTrust, Ciao, MovieLens, and Netflix. The number of ratings in these datasets ranges from tens of thousands to several million. The authors chose the precision and recall scores as metrics of evaluation because their design purpose is not to predict ratings, but to provide a ranked list of items. In experiments, the authors compared the performance of the two techniques on dimensionality reduction and found that PCA is preferable. They further studied two cases of adding constraints in experiments. One of them is to investigate the impact of including neighbors on the precision and recall scores measured with the cosine similarity. The other is to penalize popular items for preserving diversity. As reported, including neighbors may increase both precision and recall scores in almost all cases. However, the improvements are mostly insignificant. The experiments also indicate that penalizing popular items can increase diversity in the recommendations.

The most interesting part of this paper is probably the ways of adding constraints in experiments and the methods of evaluating performance. This can be a good reference for readers working on collaborative filtering.