A Harmonic Extension Approach for Collaborative Ranking

By Da Kuang, Zuoqiang Shi, Stanley J. Osher, Andrea L. Bertozzi. 2016

6 citations recorded by the source corpus.

We present a new perspective on graph-based methods for collaborative ranking for recommender systems. Unlike user-based or item-based methods that compute a weighted average of ratings given by the nearest neighbors, or low-rank 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 item-item 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-of-the-art methods.

Subfields: Computer Vision and Pattern Recognition, Information Systems, Signal Processing

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computer Vision and Pattern Recognition

1. Michael M. Bronstein
2. Luc Van Gool
3. Dacheng Tao
4. Xiaogang Wang
5. Jitendra Malik
6. Ming-Hsuan Yang
7. Trevor Darrell
8. Andrea Vedaldi

In Information Systems

1. Yann LeCun
2. Pierre Vandergheynst
3. Michael M. Bronstein
4. Federico Monti
5. Xavier Bresson
6. Arthur Szlam

In Signal Processing

1. Federico Monti
2. Xavier Bresson
3. Michael M. Bronstein
4. Joan Bruna
5. Yann LeCun
6. Pierre Vandergheynst
7. Arthur Szlam
8. Yoshua Bengio

Citations in the corpus

• Geometric Matrix Completion with Recurrent Multi-Graph Neural Networks (2017, background): “Recently, there have been several attempts to incorporate geometric structure into matrix completion problems (Ma et al., 2011; Kalofolias et al., 2014; Rao et al., 2015; Kuang et al., 2016), e.g. in the form of column and row graphs representing similarity of users and items, respectively.”
• Geometric Deep Learning: Going beyond Euclidean data (2016, background): “structure into matrix completion problems [94], [95], [96], [97] in the form of column- and row graphs representing similarity of users and items, respectively (see Figure 4).”