Matrix Completion on Graphs

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, Candès 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.

Matrix Decomposition on Graphs: A Functional View

“…some geometric structure of the matrix rows and columns. There exist several prior works on geometric matrix completion that exploit geometric information (Berg et al, 2017;Kalofolias et al, 2014;Rao et al, 2015) such as graphs encoding relations between rows and columns. More recent works leverage deep learning on geometric domains (Berg et al, 2017;Monti et al, 2017) to extract relevant information from geo-metric data such as graphs.…”

“…For a fair comparison with (Boyarski et al, 2020), we use graphs taken from the synthetic Netflix dataset. Synthetic Netflix is a small synthetic dataset constructed by (Kalofolias et al, 2014) and (Monti et al, 2017), in which the user and item graphs have strong community structure. Noise Ours Ours-FM SGMC 5 1e-3 2e-3 5e-3 10 4e-3 3e-3 1e-2 20 6e-3 6e-3 1e-2…”

“…Flixster ML-100K MC (Candès & Recht, 2009) 1.533 0.973 GMC (Kalofolias et al, 2014) -0.996 GRALS (Rao et al, 2015) 1.245 0.945 RGCNN (Monti et al, 2017) 0.926 0.929 GC-MC (Berg et al, 2017) 0.917 0.910 Ours-FM 1.02 1.12 DMF (Arora et al, 2019) 1 In addition to synthetic Netflix, we also validate our method on two more recommender systems datasets for which row and column graphs are available. Movielens-100K (Harper & Konstan, 2016) contains ratings of 1682 items by 943 users whereas Flixter (Jamali & Ester, 2010) contains ratings of 3000 items by 3000 users.…”

“…Often, the low-dimensional manifold information is exploited via a graph structure between the data samples. As a result, graphs are often used as regularizers in various machine learning problems such as dimensionality reduction (Jiang et al, 2013), hashing (Liu et al, 2011) or matrix completion (Kalofolias et al, 2014) to name a few. In this article, we focus on dimensionality reduction and matrix completion and propose a principled framework that gives a unified solution to both problems by modelling the extra geometric information available in terms of graphs.…”

“…Various problems in collaborative filtering can be posed as a matrix completion problem (Kalofolias et al, 2014;Rao et al, 2015), where for example the columns and rows represent users and items, respectively, and matrix values represent a score determining the preference of user for that item. Often, additional structural information is available in the form of column and row graphs representing similarity of users and items, respectively.…”