A Harmonic Extension Approach for Collaborative Ranking

Da Kuang UCLA Zuoqiang Shi Tsinghua University Stanley Osher UCLA A. Bertozzi UCLA

Information Theory Numerical Analysis and Scientific Computing mathscidoc:1711.19001

International Symposium on Nonlinear Theory & Its Applications (NOLTA), 2017
We present a new perspective on graph-based methods for collaborative ranking for recommender systems. Unlike user-based or itembased methods that compute a weighted average of ratings given by the nearest neighbors, or lowrank approximation methods using convex optimization and the nuclear norm, we formulate matrix completion as a series of semi-supervised learning problems, and propagate the known ratings to the missing ones on the user-user or itemitem graph globally. The semi-supervised learning problems are expressed as Laplace-Beltrami equations on a manifold, or namely, harmonic extension, and can be discretized by a point integral method. We show that our approach does not impose a low-rank Euclidean subspace on the data points, but instead minimizes the dimension of the underlying manifold. Our method, named LDM (low dimensional manifold), turns out to be particularly effective in generating rankings of items, showing decent computational efficiency and robust ranking quality compared to state-ofthe- art methods.
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@inproceedings{da2017a,
  title={A Harmonic Extension Approach for Collaborative Ranking},
  author={Da Kuang, Zuoqiang Shi, Stanley Osher, and A. Bertozzi},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20171107104059565966850},
  booktitle={International Symposium on Nonlinear Theory & Its Applications (NOLTA)},
  year={2017},
}
Da Kuang, Zuoqiang Shi, Stanley Osher, and A. Bertozzi. A Harmonic Extension Approach for Collaborative Ranking. 2017. In International Symposium on Nonlinear Theory & Its Applications (NOLTA). http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20171107104059565966850.
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