Matrix Completion on Graphs

TL;DR

This paper introduces a graph-structured matrix completion model that leverages community information and manifold learning to improve recovery of missing data in matrices, especially in real-world scenarios like recommender systems.

Contribution

It proposes a novel convex optimization framework that incorporates row and column proximities via graphs, enhancing matrix completion performance over traditional low-rank methods.

Findings

Outperforms standard matrix completion in synthetic and real datasets.

Effectively captures community structures in data.

Provides a convergent iterative solution scheme.

Abstract

The problem of finding the missing values of a matrix given a few of its entries, called matrix completion, has gathered a lot of attention in the recent years. Although the problem under the standard low rank assumption is NP-hard, Cand\`es and Recht showed that it can be exactly relaxed if the number of observed entries is sufficiently large. In this work, we introduce a novel matrix completion model that makes use of proximity information about rows and columns by assuming they form communities. This assumption makes sense in several real-world problems like in recommender systems, where there are communities of people sharing preferences, while products form clusters that receive similar ratings. Our main goal is thus to find a low-rank solution that is structured by the proximities of rows and columns encoded by graphs. We borrow ideas from manifold learning to constrain our solution to be smooth on these graphs, in order to implicitly force row and column proximities. Our matrix recovery model is formulated as a convex non-smooth optimization problem, for which a well-posed iterative scheme is provided. We study and evaluate the proposed matrix completion on synthetic and real data, showing that the proposed structured low-rank recovery model outperforms the standard matrix completion model in many situations.