  As an important tool for information filtering in the era of socialized web,
recommender systems have witnessed rapid development in the last decade. As
benefited from the better interpretability, neighborhood-based collaborative
filtering techniques, such as item-based collaborative filtering adopted by
Amazon, have gained a great success in many practical recommender systems.
However, the neighborhood-based collaborative filtering method suffers from the
rating bound problem, i.e., the rating on a target item that this method
estimates is bounded by the observed ratings of its all neighboring items.
Therefore, it cannot accurately estimate the unobserved rating on a target
item, if its ground truth rating is actually higher (lower) than the highest
(lowest) rating over all items in its neighborhood. In this paper, we address
this problem by formalizing rating estimation as a task of recovering a scalar
rating function. With a linearity assumption, we infer all the ratings by
optimizing the low-order norm, e.g., the $l_1/2$-norm, of the second derivative
of the target scalar function, while remaining its observed ratings unchanged.
Experimental results on three real datasets, namely Douban, Goodreads and
MovieLens, demonstrate that the proposed approach can well overcome the rating
bound problem. Particularly, it can significantly improve the accuracy of
rating estimation by 37% than the conventional neighborhood-based methods.
