Paper:
Mixture-Rank Matrix Approximation for Collaborative Filtering
Low-rank matrix approximation (LRMA) methods have achieved excellent accuracy among today's collaborative filtering (CF) methods. In existing LRMA methods, the rank of user/item feature matrices is typically fixed, i.e., the same rank is adopted to describe all users/items. However, our studies show that submatrices with different ranks could coexist in the same user-item rating matrix, so that approximations with fixed ranks cannot perfectly describe the internal structures of the rating matrix, therefore leading to inferior recommendation accuracy. In this paper, a mixture-rank matrix approximation (MRMA) method is proposed, in which user-item ratings can be characterized by a mixture of LRMA models with different ranks. Meanwhile, a learning algorithm capitalizing on iterated condition modes is proposed to tackle the non-convex optimization problem pertaining to MRMA. Experimental studies on MovieLens and Netflix datasets demonstrate that MRMA can outperform six state-of-the-art LRMA-based CF methods in terms of recommendation accuracy.

Review:
This is an excellent paper, proposing a sound idea of approximating a partially defined rating matrix with a combination of multiple low rank matrices of different ranks in order to learn well the head user/item pairs (users and items with lots of ratings) as well as the tail user/item pairs (users and items we few ratings). The idea is introduced clearly. The paper makes a good review of the state-of-the-art, and the experiment section is solid with very convincing results.
In reading the introduction, the reader could find controversial the statement in lines 25-27 about the correlation between the number of user-item ratings and the desired rank. One could imagine that a subgroup of users and items have a large number of ratings but in a consistent way, which can be explained with a low rank matrix. The idea is getting clear further in the paper, when explained in the light of overfitting and underfitting. The ambiguity could be avoided in this early section by adding a comment along the line of “seeking a low rank is a form of regularization”.