a New Approach to Collaborative Filtering: Operator Estimation with Spectral Regularization

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Introduction

Collaborative filtering (CF) aims to predict preferences of a set of users for objects such as movies, music, or any object based on previously purchased or rated objects by these users. The goal is to offer new objects to the users based on the predicted preferences.

Regularization-based CF methods have recently been attracting <ref name="srebro03">N. Srebro and T. Jaakkola. Weighted low-rank approximations. In T. Fawcett and N.Mishra, editors, Proceedings of the Twentieth International Conference on Machine Learning, pages 720–727. AAAI Press, 2003. </ref>. These methods only incorporate the users preferences which can be given as a preference matrix in which the rows and columns represent users and objects, respectively. The problem will be inferring the unknown entries based on the known ones.

In order to solve the problem, a low-rank matrix should be found that approximates the partially observed given preference matrix. This rank constraint is a regularization. Heuristics <ref name="srebro03"> </ref> and penalizing the predicted matrix by its trace norm <ref>N. Srebro, J. D. M. Rennie, and T. S. Jaakkola. Maximum-margin matrix factorization. In L. K. Saul, Y. Weiss, and L. Bottou, editors, Adv. Neural. Inform. Process Syst. 17, pages 1329–1336, Cambridge, MA, 2005. MIT Press. </ref> are two approaches to solve the resulted optimziation problem.

Abernethy et al.<ref name="self"> J. Abernethy, F. Bach, T. Evgeniou, J. P. Vert. A New Approach to Collaborative Filtering: Operator Estimation with Spectral Regularization. Journal of Machine Learning Research, Vol. 10(Mar):803--826, 2009</ref> present a generalized regularization-based approach for CF which learns the linear operators mapping users space to their desired objects space.

The previledge and novelty of this approach is the ability to incorporate additional information, such as different attributes of users/objects by means of kernel methods. These attributes can help through predicting the unkown entries of the preference matrix. They also show that some multi-tasking learning methods are special cases of their general framework.



References

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