  The importance of accurate recommender systems has been widely recognized by
academia and industry. However, the recommendation quality is still rather low.
Recently, a linear sparse and low-rank representation of the user-item matrix
has been applied to produce Top-N recommendations. This approach uses the
nuclear norm as a convex relaxation for the rank function and has achieved
better recommendation accuracy than the state-of-the-art methods. In the past
several years, solving rank minimization problems by leveraging nonconvex
relaxations has received increasing attention. Some empirical results
demonstrate that it can provide a better approximation to original problems
than convex relaxation. In this paper, we propose a novel rank approximation to
enhance the performance of Top-N recommendation systems, where the
approximation error is controllable. Experimental results on real data show
that the proposed rank approximation improves the Top-$N$ recommendation
accuracy substantially.
