Inductive Matrix Completion Based on Graph Neural Networks
Authors: Muhan Zhang
Yixin Chen
Venue:  (2020)
Keywords: matrix completion
graph neural network

Abstract: We propose an inductive matrix completion model without using side information. By factorizing the (rating) matrix into the product of low-dimensional latent embeddings of rows (users) and columns (items), a majority of existing matrix completion methods are transductive, since the learned embeddings cannot generalize to unseen rows/columns or to new matrices. To make matrix completion inductive, most previous works use content (side information), such as user's age or movie's genre, to make predictions. However, high-quality content is not always available, and can be hard to extract. Under the extreme setting where not any side information is available other than the matrix to complete, can we still learn an inductive matrix completion model? In this paper, we propose an Inductive Graph-based Matrix Completion (IGMC) model to address this problem. IGMC trains a graph neural network (GNN) based purely on 1-hop subgraphs around (user, item) pairs generated from the rating matrix and maps these subgraphs to their corresponding ratings. It achieves highly competitive performance with state-of-the-art transductive baselines. In addition, IGMC is inductive -- it can generalize to users/items unseen during the training (given that their interactions exist), and can even transfer to new tasks. Our transfer learning experiments show that a model trained out of the MovieLens dataset can be directly used to predict Douban movie ratings with surprisingly good performance. Our work demonstrates that: 1) it is possible to train inductive matrix completion models without using side information while achieving similar or better performances than state-of-the-art transductive methods; 2) local graph patterns around a (user, item) pair are effective predictors of the rating this user gives to the item; and 3) Long-range dependencies might not be necessary for modeling recommender systems.

Published as a conference paper at ICLR 2020
INDUCTIVE MATRIX COMPLETION BASED ON GRAPH NEURAL NETWORKS
Muhan Zhang Washington University in St. Louis
Yixin Chen Washington University in St. Louis chenêcse. wustl. edu
muhan@wustl edd *Now at Facebook
ABSTRACT
We propose an inductive matrix completion model without using side information. By factorizing the (rating) matrix into the product of low-dimensional latent em- beddings of rows (users) and columns (items), a majority of existing matrix com- pletion methods are transductive, since the learned embeddings cannot generalize to unseen rows/columns or to new matrices. To make matrix completion inductive, most previous works content (side information), such user's age movie's s genre, to make predictions. However, high-quality content is not always available, and can be hard to extract. Under the extreme setting where not any side informa- tion is available other than the matrix to complete, can we still learn an inductive matrix completion model? In this paper, we propose Inductive Graph-based Matrix Completion (IGMC) model to address this problem. IGMC trains a graph neural network (GNN) based purely on 1-hop subgraphs around (user, item) pairs generated from the rating matrix and maps these subgraphs to their corresponding ratings. achieves highly competitive performance with state-of-the-an transduc- tive baselines. In addition, IGMC is inductive it can generalize to users/items unseen during the training (given that their interactions exist), and can even trans- fer to new tasks. Our transfer learning experiments show that model trained out the MovieLens dataset can be directly used to predict Douban movie ratings with surprisingly good performance. Our work demonstrates that: 1) it is possible to train inductive matrix completion models without using side information while achieving similar better performances than state-of-the-arr transductive meth- ods; 2) local graph patterns around a (user, item) pair are effective predictors the rating this user gives to the item; and 3) Long-range dependencies might not be necessary for modeling recommender systems.
INTRODUCTION
Matrix completion (Candès & Recht 2009) is one common formulation of recommender systems, where rows and columns of matrix represent interest in items corresponds to filling in the missing entries of the rating matrix. By assuming a low-rank rating matrix, many of the most popular matrix completion algorithms use factorization techniques that decompose rating rij into w; h,. the inner product of user i's and item j's la- tent feature vectors w, and h,, respectively, which have achieved great successes (Adomavicius & Tuzhilin 2005 Schafer et al. [007 Koren et al., 2009 Bobadilla et al. 2013)
and items, respectively, and predicting users'
However, matrix factorization is intrinsically transductive, meaning that the learned latent features (embeddings) for users/items are not generalizable to users/items unseen during the training. When the rating matrix has changed values or has new rows/columns added, often requires a complete retraining to get the new embeddings. To make matrix completion inductive, Inductive Matrix Com- pletion (IMC) has been proposed, which leverages content (side information) of users and items (Jain) & Dhillon 2013 Xu et al. 2013).. In IMC, rating is decomposed by rú x, Qy y. where and y, are content feature vectors of user and item j. respectively, and Q is a learnable matrix modeling the feature interactions. To accurately predict missing entries, IMC methods have strong constraints on the content quality, which often leads to inferior performance when high-quality content is not available. Other content-based recommender systems (Lops al. 2011) face similar problems. In

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Published as a conference paper at ICLR 2020
Users
Items
Graph neural network
Items
user everage rsting -3
2
5
item tobal miings morived =2 similar users' average miinng
item average rating 55
(like)
L
# ( :0:000000000 +2 -
#u
-1 =1
Users
5
Extract enclosing subgraphs
Learn graph patters
Predict ratings
S
user everage rating -4 item avenge rabing -1 Item tocal ratings rocelved similar users' average of (0 -,.0 +,
5
5
(dislike)
Figure 1: We extract local enclosing subgraph around each rating (dark green), and train GNN to map subgraphs to ratings. Each enclosing subgraph is induced by the user and item associated with the target rating well their h-hop neighbors (here h= 1). From the subgraphs, GNN can learn mixed graph patterns (such as average ratings, paths, etc.) useful for rating prediction. We illustrate some possible patterns in the GNN box. We use the trained GNN complete the missing entries.
some extreme settings, there is not any content available for user, such as a website where users are completely anonymous. In these cases, inductive matrix completion seems impossible.
Recently, |Harffrd et al. (2188) propose exchangeable matrix layers, which apply permutation equiv- ariant operations on the rating matrix reconstruct missing entries. The resulting models are in- ductive and do not rely on side information. However, these models take the whole rating matrix R input and output another reconstructed matrix, which poses scalability challenges for practical datasets with millions of users/items. In this paper, we propose a novel inductive matrix completion method does not need take the whole rating
method that does not use any content. Further, matrix as input, and can infer ratings for individual user-item pairs. The key that frees us from using content or whole rating matrix is local graph pattern. If for each observed rating we add an edge between the corresponding user and item, we can build bipartite graph from the rating matrix. Subsequently, predicting unknown ratings converts equivalently to predicting labeled links in this bipartite graph. This transforms matrix completion into link prediction problem -owell & Kleinberg 2007), where graph patterns play a major role in determining link existences.
A major class of link prediction methods are heuristic methods, which predict links based on some heuristic scores. For example, the common neighbors heuristic count the common neighbors be- tween two nodes to predict links, while the Katz index (Katz) 1953) uses weighted sum of all the walks between two nodes. See (Liben-Nowell Kleinberg 2007 for an overview. These heuristics can be seen as some predefined graph structure features calculated based on the local global graph patterns around links, which have achieved great successes due to their simplicity and effectiveness.
However, these traditional link prediction heuristics only work for simple graphs where nodes and edges both only have a single type. Can we find some heuristics for labeled link prediction in bipartite graph? Intuitively, such heuristics should exist. For example, if
uo likes item to. we may expect to see very often that Up is also liked by some other user un who shares a similar taste ug- By similar taste, we mean 2j and un have together both liked some other item V1. In the bipartite graph, such a pattern is realized as a "like" path (uo -hiike U1 -tiked by -tikke vo). If there are many such paths between Uy and Vo, we may infer that uo highly likely to like Vo. Thus, we may count the number of such paths as an indicator of how likely uo likes up. In fact, many neighborhood based recommender systems (Desrosiers Karypis 2011) rely on similar heuristics.
Of course we can try to manually define many such intuitive heuristics and test their effectiveness. In this work, however, we take different approach that automatically learns suitable heuristics from the given bipartite graph. To do so, we first extract an h-hop enclosing subgraph for each training user-item pair (u, v), which is defined to be the subgraph induced from the bipartite graph by nodes u, and their neighbors within h hops. Such local subgraphs contain rich graph pattern information about the rating that u may give to V. For example, all the (uo -Like U/ >iiked by U1 ~Like Un) paths included in the h-hop enclosing subgraph around (wo, vg). By feeding these enclosing subgraphs
to graph neural network (GNN), we train graph regression model that maps each subgraph to the rating that its center user gives to its center item. Figure IT illustraaee the overall framework.
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Due to the superior graph learning ability, a GNN can learn highly expressive graph structure fea- tures useful for inferring the ratings without restricting the features to predefined heuristics. Given a trained GNN, can also apply it to unseen users/items without retraining. Our resulting algorithm is inductive, and is named Inductive Graph-based Matrix Completion (IGMC). Note that IGMC does not address the extreme cold-start problem, as it still requires unseen user-item pair's en- closing subgraph (i.e., the user and item should at least have some interactions with neighbors so that the enclosing subgraph is not empty). This scenario is very common in practice. For example, a newly registered YouTube user may quickly watch some videos without completing their personal information. In this case, if we cannot retrain user embeddings frequently. IGMC can be of great value by still making recommendations based purely on this user's interaction history with videos.
We compare IGMC with state-of-the-arr matrix completion algorithms on five benchmark datasets. Without using any content, IGMC achieves the smallest RMSEs on four of them, even beating many transductive baselines augmented by side information. Our model is also equipped with excellent transfer learning ability. We show that IGMC model trained on the MovieLens-100k dataset can be directly used to predict Douban movie ratings and even outperforms some baselines trained specifically Douban. We also analyze IGMC's behavior sparse rating matrices. We show that IGMC is more robust than transductive methods on spanse matrices. Under an extremely sparse case (only 0.1 % of Moviel.ens- IM training ratings are kept), IGMC can still achieve less than 0.95 RMSE, beating a state-of-the-ard transductive method GC-MC by more than 0. RMSE. Finally, our visualization confirms that local enclosing subgraphs are indeed strong predictors of ratings.
2 RELATED WORK
Graph neural networks Graph neural networks (GNNs)
new type of neural networks for
learning over graphs (Scarselli et al. 2009;
Bruna et al. Duvenaud et al. 2015 Li et al. 2015
2013
Kipf & Welling Niepert et al. 2016: Dai et al. 2016). There are two types of GNNs; Node-
level GNNs use message passing layers to iteratively pass messages between each node and its neighbors in order to extract feature vector for each node encoding its local substructure. Graph-
level GNNs additionally use a pooling layer such as summing which aggregates node feature vectors into a graph representation to enable graph-level tasks such as graph slasiffcation/regression. Due to oftte--f-thh-an the-art performance on oumi-supervised node classification (Kipf & Welling, 2016), network embedding (Hamilton et al.) 2017]. graph classification (Zhang et al. 2018), and link prediction (Zhang & Chen 2018), etc.
the superior graph representation learning ability, GNNs have achieved state-of
GNNs for matrix completion The matrix completion problem has been studied using GNNs. Monti et al. (2017) develop multi-graph CNN (MGCNN) model to extract user and item latent features from their respective seirest-neighbor networks. [Berg al.] (2017) propose graph convo- lutional matrix completion (GC-MC) which directly applies a GNN to the user-item bipartite graph to extract user and item latent features using a GNN. The SpectralCT model of (Zheng et al.) 2018) uses speetral- GNN on the bipartite graph to learn node embeddings. Although using GNNs for matrix completion, all these models are still transductive- - MGCNN and SpectralCF require graph Laplacians which do not generalize new graphs, while GC-MC uses one-hot encoding of node IDs as initial node features, thus cannot generalize to unseen users/items. A recent inductive graph- based recommender system, PinSage (Ying et al. 2018a),, uses node content as initial node features (instead of the one-hot encoding in GC-MC), and is successfully used in recommending related pins in Pinterest. Although being inductive, PinSage relies heavily on the rich visual and text content associated with the pins which is not often accessible in other recommendation tasks. In compari- son, our IGMC model inductive and does not rely on any content. All previous approaches use node-level GNNs to learn embeddings for nodes, while our IGMC uses graph-level GNN to learn representations for subgraphs. We will discuss this erucial difference in more details in Section
Another related previous work is (Hartford et al.) 2018), which defines exchangeable matrix layers to perform permutation--equiaaiise
operations on matrices to achieve inductive matrix completion without using content. In particular, the operation updates each matrix entry by a weighted sum of itself, entries of its row, entries of its column, and all other entries of the matrix, where parameters for each of the four components are shared across all entries. It can also be regarded the exceptions that 1) the message passing is performed on edges (final edge features are pooled into node features), and 2) all edges (including those not connected the center edge) pass messages
GNN with
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Algorithm ENCLOSING SUBGRAPH EXTRACTION
input: h, target user-item pair (u, v), the bipartite graph
2: output: the ft-hop enclosing subgraph G for (u, v)
3: U Upringe {u} , V-
Viringe {w}
4: for i = .......,, do
5:
Uninge {u+ : u4 Viringe) \
6: V Fringe for :44 Upringe) V
7:
Upinge Upringe: Vyringe V Viinge
8:
U- U U Ugringe V= V J Vringe
9: Let G be the vertex-induced subgraph from G using vertices U, V
Remove edge (u, v)) from G
11: end for
12: return Gh
Note:
{ur ui Vysige is the set of nodes that are adjacent to at least one node in Vising with any edge type.
to the center edge in each round. One limitation of (Hartford et al. 2018) is that it takes the entire rating matrix as input, which might raise concerns for large matrices. In comparison, our IGMC takes only local subgraphs input which avoids the issue and enables predicting individual ratings.
Link prediction based on graph patterns Learning supervised heuristics (graph patterns) has been studied for link prediction in simple graphs. Zhang & Chen (2017) propose Weisfeiler-Lehmanl Neural Machine (WLNM), which learns graph structure features using a fully-connected neural network on the subgraphs' adjacency matrices. Later, they improve this work by replacing the fully- connected neural network with GNN and achieves state-of-the-art link prediction results (Zhang & Chen 2018). Our work generalizes this line of research from predicting link existence in simple graphs to predicting values of links in bipartite graphs (i.e., matrix completion). In (Chen et al. 2005 Zhou al. 2007). traditional link prediction heuristics are adapted to bipartite graphs which show promising performance for recommender systems. Our work differs in that we do not use any predefined heuristics, but learn general graph structure features using GNN. Another similar work to ours is (Li & Chen 2013), where graph kernels are used to learn graph structure features. However, graph kernels require quadratic time and space complexity to compute and store the kernel matrices thus unsuitable for modern recommender systems.
INDUCTIVE GRAPH-BASED MATRIX COMPLETION (IGMC)
We now present Inductive Graph-based Matrix Completion (IGMC) framework. We use G to denote the undirected bipartite graph constructed from the given rating matrix R. In G, a node is either (denoted by u, corresponding
row in R) an item (denoted by u, corresponding to a column in R). Edges can exist between user and item, but cannot exist between two users or two the rating that u gives v. We use R to denote the set of all possible ratings (e.g., R = {1, 2, 3, 4, 5} in MovieLens), and use (u) to denote the set of u's neighbors that connect with edge type r.
items. Each edge (u, v) has value R,,,, corresponding
3.1 ENCLOSING SUBGRAPII EXTRACTION
The first part of IGMC is enclosing subgraph extraction. For each observed rating Ru,,, we extract an h-hop enclosing subgraph around (u, v) from G. Algorithm describes the BFS procedure for extracting hi-hop enclosing subgraphs. We will feed these enclosing subgraphs a GNN and regress on their ratings. Then, for each testing (u, v) pair, we again extract its h-hop enclosing subgraph from G. and use the trained GNN model to predict its rating. Note that after extracting training enclosing subgraph for (u, v), we should remove the edge (u, v) because it is the target to predict.
3.2 NODE LABELING
The second part of IGMC is node labeling. Before we feed an enclosing subgraph to the GNN, we first apply node labeling it, which gives integer label to every node in the subgraph. The
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purpose is to use different labels to mark nodes' different roles in subgraph. Ideally, our node labeling should be able to: 1) distinguish the target user and target item between which the target rating is located, and 2) differentiate user-type nodes from item-type nodes. Otherwise, the GNN cannot tell between which user and item to predict the rating, and might lose node-type information. To satisfy these conditions, we propose a node labeling follows: We first give label 0 and 1 to the target user and target item, respectively. Then, we determine other nodes' labels according to at which hop they are included in the subgraph in Algorithm I If a user-type node is included at the jt hop, we will give it a label 2i. If an item-typo node is included at the 2th hop, we will give it 2i + 1. Such node labeling can sufficiently discriminate: 1) target nodes from "context" nodes, 2) users from items (users always have even labels), and 3) nodes of different distances to the target rating.
Note that this not the only possible way of node labeling, but we empirically verified its excellent performance. The one-hot encoding of these node labels will be treated as the initial node features x" of the subgraph when fed the GNN. Note that our node labels are determined completely inside each enclosing subgraph, thus are independent of the global bipartite graph. Given new enclosing subgraph, we can as well predict its rating even all of its nodes are from a different bipartite graph, because IGMC purely relies on graph patterns within local enclosing subgraphs without leveraging any global information specific to the bipartite graph. Our node labeling is also different from using the global node IDs as in GC-MC (Berg et al. (017). Using one-hot encoding of global IDs is essentially transforming the first message passing layer's parameters into latent node embedding associated with each particular ID (equivalent an embedding lookup table). Such model cannot generalize to nodes whose IDs are out of range, thus is transductive.
3.3 GRAPH NEURAL NETWORK ARCHITECTURE
The third part of IGMC is to train a graph neural network (GNN) model predicting ratings from the enclosing subgraphs. In previous node-based approaches such as GC-MC, a node-level GNN is applied to the entire bipartite graph to extract node embeddings. Then, the node embeddings of and are input to an maner-produe or bilinear operator to reconstruct the rating on (u, v). In contrast, IGMC applies graph-level GNN to the enclosing subgraph around (u, v) and maps the subgraph to the rating. There are thus two components in our GNN: 1) message passing layers that extract feature vector for each node in the subgraph, and 2) pooling layer summarize subgraph representation from node features.
To learn the rich graph patterns introduced by the different edge types, we adopt the relational graph convolutional operator (R-GCN) (Schlichtkrull et al. 2018) as our GNN's message passing layers, which has the following form:
++
W X
+ Σ Σ (.((]] W' FER JEN-(0)
(I)
where x denotes node i's feature vector at layer 1, W and { (W',Ir E R) are learnable parameter matrices. Since neighbors connected to with different edge types r' are processed by different parameter matrices W we are able learn a large amount of enriched graph patterns inside the edge types, such as the average rating the target user gives to items, the average rating the target item receives, and by which paths the two target nodes are connected, etc. We stack L message passing layers with tanh activations between two layers. Following (Zhang et al.] 2018]] Xu et al. 2018) node i's feature vectors from different layers are concatenated as its final representation h;;
h concat(x),
Next, we pool the node representations into graph-level feature vector. There are many choices such as summing, averaging, SortPooling (Zhang et al., 2018), DiffPooling (Ying et al.) 2018b). etc. In this work, however, we use different pooling layer which concatenates the final representations of only the target user and item as the graph representation:
g = concat(h,, h.):
where we use h, and h, to denote the final representations of the target user and target item, re- spectively. Our particular choice is due to the extra importance that these two target nodes carry
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compared other context nodes. Although being very simple, empirically verified its better performance than summing and other advanced pooling layers for our matrix completion tasks.
After getting the final graph representation,
MLP to output the predicted rating:
w σ (Wg),
(4)
where W and are parameters of the MLP which map the graph representation to scalar rating and a is an activation function (we take ReLU in this paper).
3.4 MODEL TRAINING
Loss function We minimize the mean squared error (MSE) between the predictions and the ground truth ratings:
C = [(u, u) Du.
Σ
(Ru,v - Ru,,)²
(5)
TH
(u,0):f....
where we use Ru, and Raw to denote the true rating and predicted rating of (u, v), repsectively, and 52 a 0/1 mask matrix indicating the observed entries of the rating matrix R.
Adjacent rating regularization The R-GCN layer ( used in our GNN has different parameters W, for different rating types. One drawback here is that it fails take the magnitude of ratings into consideration. For instance, a rating of 4 and rating of in MovieLens both indicate that the user likes the movie, while a rating of indicates that the user does not like the movie. Ideally, we expect our model to be aware of the fact that rating of 4 is more similar to 5 than is. In R- GCN, however, ratings 1, 4 and 5 are only treated as three independent edge types the magnitude and order information of the ratings is completely lost. To fix that, we propose adjacent rating regularization (ARR) technique, which encourages ratings adjacent to each other to have similar parameter matrices. Assume the ratings in R exhibit ordering T1, "1,72,..... F|R) which indicates increasingly higher preference that users have for items. Then, the ARR regularizer is:
LARR
Σ 2 R|-1
W r: - Wr:
where ||=||æ denotes the Frobenius norm of matrix. The above regularizer restrains the parameter matrices adjacent ratings from having too much differences, which not only takes into considera- tion of the ratings' order, but also helps the optimization of those infrequent ratings by transferring knowledge from their adjacent ratings. The final loss function is given by:
Linal C XLARR.
where À trades-off the importance of the MSE loss and the ARR regularizer. There are many other ways to model rating magnitude and order, which are left for future work.
4 GRAPH-LEVEL GNN VS. NODE-LEVEL GNN
Compared to previous graph matrix completion ap- proaches such as PinSage and GC-MC, impor- tant difference of IGMC is that it uses a graph-level GNN map the enclosing subgraph around the tar- get user and item to their rating (left figure (a)), in- stead of using node-level GNN on the bipartite graph G to learn target user's and item's embed- dings and use the node embeddings to predict the rating (left figure (b)). One drawback of the latter node-based approach is that the learned node embed- dings are essentially encoding the two rooted sub- trees around the two nodes independently, which fails to model the interactions and correspon- dences between the nodes of the two trees. For example, from the two subtrees of the left figure (b) we do not really know whether the two target nodes are just isolated from each other like in (b) or actually densely connected like in (a); these two cases look identical to node-based approach.
Users
Items
Users
Items
B
5
2
3
Subgraph emboodiing
Subtree embedding
Subtree emhedding
Rating
(a)
Rating
(b)

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Table 1: Statistics of each dataset.
Dataset
Users
Items
Ratings
Density
Rating types
Flixster
3,000
3,000
26,173
0.0029
0.5, I, 1.5, ... 5
Douban
3,000
3,000
136,891
0.0152
1,
YahooMusic
3,000
3,000
5,335
0.0006
1,2,3, 100
ML-100K
943
1,682
100,000
0.0630
1,2,3,4,
ML-IM
6,040
3,706
1,000,209
0.0447
3,23,445
In comparison, a graph-level GNN can discriminate the two cases through a sufficient number of message passing rounds. Since the learning is confined to the subgraph, stacking multiple graph convolution layers will learn more and more refined local structural features which can adequately discriminate up to all subgraphs that the Weisfeiler- Lehman algorithm can discriminate (Xu et al. 2018). However, for node-based approaches, since there is no subgraph boundary, stacking multi- ple graph convolutions will only extend the convolution range to unrelated distant nodes and over- smooth the node embeddings (Li et al.] 2018). This is reflected in that previous node-bacd based ap- proaches mainly use only one or message passing layers (Berg al., 2017; Ying al., 2018a).
Nevertheless, using a graph-level GNN on every target rating's enclosing subgraph has higher com- plexity than using node-level GNN the entire bipartite graph. Suppose the bipartite graph has [E] edges. Then performing one round of message passing using a node-level GNN has O(EE) complexity. Assume the maximum number of edges in all enclosing subgraphs K. Perform- ing one round of message passing for all enclosing subgraphs then has O(K complexity. In practice, we can use subsampling restrict K a small number to reduce IGMC's complexity.
5 EXPERIMENTS
We conduct experiments on five common matrix completion datasets: Flixster (Jamali & Es- ter 2010), Douban (Ma et al., 2011), YahooMusic (Dror et al. 2011). MovieLens- and MovieLens-IM (Miller et al.) 2003). For ML-100K, we train and evaluate on the canonical bl.beee .test train/test split. For ML-IM, we randomly split it into 90% and 10% train/test sets. For Flixster, Douban and YahooMusic we use the preprocessed subsets and splits provided We implemented IGMC using pytorch geometric (Fey & Lenssen| 2019). We tuned model hyperparameters based cross vali- dation results on M0.-10OK, and used them across all datasets. The final architecture uses 4 R-GCN layers with 32, 32, 32, 32 hidden dimensions. Basis decomposition with 4 bases is used to reduce the number of parameters in W, (Schlichtkrull et al. 2018). The final MLP has 1 128 hidden units and dropout rate of 0.5. We use I-hop enclosing subgraphs for all datasets, and find them sufficiently good. We find using 2 or more hops can slightly increase the performance but take much longer training time. For each enclosing subgraph, we randomly drop out its adjacency matrix entries with probability of 0.2 during the training. We set the X in (7) to 0.001. We train our model using the Adam optimizer (Kingma & Bal (0144) with a batch size of and an initial learning rate of 0.001, and multiply the learning rate by 0.1 every 20 epochs for MI M, and every 50 epochs for all other datasets. Our code is publicly available at https: / com/ htm//uuanzzaang IGMC
by (Monti et al. 2017). Dataset statistics are summarized in Table [
5.1 FLIXSTER, DOUBAN AND YAHOOMUSIC
For these three datasets, we compare our IGMC with GRALS (Rao et al. 2015). sRGCNN (Monti et al. 2017), GC-MC (Berg et al., 2017). F-EAE (Hartford et al. 2018). and PinSage (Ying et al. 2018a). Among them, GRALS is graph regularized matrix completion algorithm. GC-MC and SRGCNN are transductive Lode-level-GNN-beee changeable matrix layers to perform inductive matrix completion without using content. PinSage is an inductive pode-level-GNN' based model using content, which is originally used to predict related pins and is adapted to predicting ratings here. We further implemented an inductive GC-MC model (IGC-MC) which replaces the one-hot encoding of node IDs with the content features to make it
matrix completion methods. F-EAE uses ex-

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Table 2: RMSE results on Flixster, Douban and YahooMusic.
Model
Inductive
Content
Flixster
Douban
YahooMusic
GRALS
no yes 1.245
0,833
38.0
sRGCNN
no
yes
0,926
0,801
22.4
GC-MC
no yes 0.917
0.734
20.5
IGC-MC
yes yes
0.999 0.999++0.062
0,990==0.082
21.39+0.989
F-EAE
ycs no 0,908
0.738
20.0
PinSage
yes yes
0.954=000005
0.739:10.002
22.9:10.229
IGMC (ours)
yes
no
0.872:+0.001
0.721++0.001
19.1+00.138
Table 3: RMSE test results MovieLens-100K (left) and MovieLens- 1M (right).
Model
Inductive
Content MI-100K
Model
Inductive
Content ML.-1M
MC
no
no
0.973
PMF
no no
0.883
IMC
no
yes
1.653
I-RBM
no no
0.854
GMC no
yes
0.996
NNMF
no
0.843
GRALS
no yes
0.945
I-AutoRec
no no
0.831
sRGCNN
no yes
0.929
CF-NADE
no n0 no
0.829
GC-MC
n1 yes
0,905
GC-MC
0.832
IGC-MC yes
yes
1.142
IGC-MC
yes yes
1,259
R-EAE yes
0.920
F-EAE
yes
0.860
PinSage
yes yes
0.951
PinSage
yes
0,906
IGMC
yes
no
0.905
IGMC
yes
0.857
inductive. The content in these datasets presented in the form of user and item graphs. We summarize whether each algorithm is inductive and whether it uses content in Table
We train our model for 40 epochs, and save the model parameters every 10 epochs. The final predic- tions are given by averaging the predictions from epochs 10, 20, 30 and 40. We repeat the experiment five times and report the average RMSES. The baseline results taken from (Hartford al. 2018). Table[]]shows the results. Our model achieves the smallest RMSES on all three datasets without us- ing any content, significantly outperforming all the compared baselines, regardless of whether they are transductive or inductive. Further, except F-EAE, all the baselines have used content informa- tion assist the matrix completion. This further highlights IGMC's great performance advantages without relying on content.
5.2 100 AND ML-1M
We further conduct experiments on MovieLens datasets. Side information is present for both users (age, gender, occupation, etc.) and movies (genres). For ML-100K, compare against matrix completion (MC) (Candès & Recht 2009), inductive matrix completion (IMC) (Jain & Dhillon 2013), geometric matrix completion (GMC) (Kalofolias et al.) 2014), as well as GRALS, SRGCNN, GC-MC, F-EAE and PinSage. We train IGMC for 80 epochs and report the ensemble performance of epochs 50, 60, 70 and 80. For ML-IM, besides the baselines GC-MC, F-EAE and PinSage, we further include soute-of-the-aee
algorithms including PMF (Mnih & Salakhutdinov 2008). I-RBM (Salakhutdinov et al.) 2007). NNMF (Dziugaite Roy, 2015). I-AutoRec (Sedhain et 2015) and CF-NADE (Zheng et al. 2016). We train IGMC for 40 epochs and report the ensemble performance of epochs 25, 30, 35 and 40. The experiments are repeated five times and the average results are reported in Table 3 (standard deviations are less than 0.001). As can see, IGMC achieves the best performance on 100K, in parallel with GC-MC despite that IGMC is an inductive model, while GC-MC is transductive and additionally uses content information. For ML-1M, IGMC cannot catch up with oftte-of-the--ae art transductive models such as CF-NADE and GC-MC, but outperforms other inductive models. We will analyze this dataset further in Section 5.3
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Table 4: RMSE transferring the model trained on ML-100K to Flixster, Douban and YahooMusic.
Model
Inductive
Content
Flixster
Douban
YuhooMusic
IGC-MC F-EAE
yes
no 1.290
1.144
25.7
yes
no 0.987
0.766
23.3
IGMC (ours)
yes no
0.906
0.759
20.1
5.3
SPARSE RATING MATRIX ANALYSIS
To gain insight into when inductive graph-based matrix completion is more suitable than transduc- tive methods, we compare IGMC with GC-MC on ML-IM under different sparsity levels of the rat- ing matrix. We sequentially increase the sparsity level by randomly keeping only 0.2, 0.1, 0.05, 0.01, and 0.001 of the original training ratings. Then, we train both models on the sparsified rating matrices, and evaluate on the original test set. Figure shows the results. As we can see, although IGMC falls behind GC-MC initially with full ratings, it starts to perform better after the sparsity ratio is less than 20%. The advantage becomes even greater under extremely sparse cases. This seems indicate that IGMC is better choice than transductive methods when there not a large amount of training data, which is particularly suitable for the initial rating collection phase of recommender system. It also suggests that transductive matrix completion relies more on the dense user-item interactions than inductive graph- matrix completion does.
GC-MC
IGMC
1.00
Test RMSE
0.95
0.90
0.85
0.2
0.1
0.05
0.01
0.d01
Spansity
Figure 2: ML- IM results under different spar- sity ratios.
5.4 TRANSFER LEARNING
A great advantage of an inductive model is its potential for transferring to other tasks. We conduct transfer learning experiment by applying the IGMC model trained on ML-100K to Flixster, Douban and YahooMusie. Among the three datasets, only Douban has exactly the same rating types ML- 100K (1,2,3,455). Thus for Flixster and YahooMusic, we bin their edge types into groups to before feeding into the MI-110KK model, and multiply the YahooMusic predictions by 20 account for the different scales. Despite all the compromises, the transferred IGMC model achieves excellent performance (Table 4p. We also show the transfer learning results of other two inductive models, IGC-MC and F-EAE. Note that an inductive model using content features (such as PinSage) is not transferrable, due the different feature spaces between MovieLens and the target datasets. Thus for IGC-MC, we replace its content features with node degrees. As we can see, IGMC outperforms the other two models by large margins in terms of transfer learning ability. Furthermore, the transferred IGMC even outperforms wide range of baselines trained especially on each dataset (Table 2
5.5 ABLATION STUDIES
To understand the individual contributions of some components IGMC, we conduct several abla- tion studies. In particular, we are interested in: 1) whether the proposed pooling layer in Equation 33 helps; 2) whether the proposed adjacent rating regularization (ARR) helps; and 3) whether incor- porating content can further improve IGMC's performance. We present the results in the appendix.
5.6 VISUALIZATION
Finally, we visualize 10 testing enclosing subgraphs with the highest and lowest predicted ratings for Flixster, Douban, YahooMusic, and ML-100K, respectively. in Figure As we can see, there substantially different patterns between high-score and low-score subgraphs, which is why IGMC can predict ratings merely from these subgraphs. For example, high-score subgraphs typically show
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both high user average rating and high item average rating, while low-score subgraphs often have mixed ratings from non-target users and have low user average rating.
1122 6.
sassersan
SLOO BA41
50 as
1256
3.448
433 n.o Flixster
Douban
F2a4 naod
iiaa
14200 iuan
$20
$22015.0
341
5,000
YahooMusic
ML-100K
Figure 3: Testing enclosing subgraphs for Flixster, Douban, YahooMusic, and 100K Top and bottom 5 are subgraphs with the highest and lowest predicted ratings, respectively. In each subgraph, nodes in the left are users, blue nodes the right are items; the bottom red and blue nodes are the target user and item; the predicted rating and true rating (in parenthesis) are shown underneath. We visualize edge ratings using the color map shown in the right. Higher ratings are redder.
CONCLUSION
In this paper, have proposed Inductive Graph-based Matrix Completion (IGMC). Instead of learn ing transductive latent features, IGMC learns local graph patterns related to ratings inductively based on graph neural networks. Compared previous inductive matrix completion methods, IGMC does not rely on content (side information) of users/items. We show that IGMC has highly competitive performance compared state-of-the-art baselines. In addition, IGMC transferrable to new tasks without any retraining, property much desired in those recommendation tasks having few training data. We hope IGMC can provide new idea matrix completion and recommender systems.
ACKNOWLEDGMENTS
The work supported in part by the National Science Foundation under award numbers III- 1226012 and SCH-1622678, and by the National Institute of Health under award number IR21HS024581.
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A ABLATION STUDIES
In this section, we conduct ablation experiments to study the individual contributions of some com- ponents in IGMC. In particular,
interested in: 1) whether the proposed pooling layer in Equation (3) helps: 2) whether the proposed adjacent rating regularization (ARR) in Equation (6) helps; and 3) whether incorporating content further improve IGMC's performance. Therefore, we compare the original IGMC with: 1) IGMC with SumPooling replacing the proposed pooling layer, 2) IGMC without ARR by setting À in 7 O, and 3) IGMC with content by concatenating target user and item's 's content feature vectors to the graph representation before feeding into the MLP 4 which is similar to (Berg et al.) 2017). The results are shown in Table
Table 5: RMSE test results of ablation experiments.
Model
Flixster
Douban
YahooMusic
MI-110KK
IGMC (original)
0.872:00.001
0.0001
19.1
0.00510.0001
IGMC (SumPooling)
0.879:10.002
0.729
22.9::1.102
0.933+000001
IGMC (no_ARR)
0.872:10.001
0.721=0.001
19.2=00140
0.908=0.001
IGMC (content)
0.886:10.002
0.726=1.001
19.1=0.069
0.906=00001
From Table 5 we have the following observations. Firstly, using the proposed pooling layer shows a huge improvement over a standard SumPooling. This might be because SumPooling assigns equal importance to all nodes in subgraph, which fails to distinguish the target user and item from the context nodes. This indicates that pooling layer able to highlight the target user and item is important for IGMC.
Secondly, we can see that disabling ARR results in 0.003 performance drop on ML-100K, while seeming have influence on the other three datasets. One possible explanation is that ARR is more useful for modeling large and dense enclosing subgraphs, since ARR enhances the modeling power by modeling the extra relationships between edge types terms of their magnitude differ- ences. Another possible reason is that we only tuned ARR's À on ML-100K and used X = 0.001 uniformly on all datasets. This partly verified by that when increasing 0.1 for YahooMusic, we can further decrease IGMC's RMSE to 18.98=0.1144 We did not fully verify this, and leave better ways of modeling rating magnitude for future study.
Thirdly, we observe that incorporating content does not improve IGMC's performance on 1L01 and often hurts the performance on the other datasets. For Flixster, Douban and YahooMusic, this phenomenon can be explained by that the content of these three datasets are user/item's 's respective graphs, which has little to no gains to model that has already exploited graph structure informa- tion between users and items very well. In addition, the content feature vectors are presented in gp00-dimensional adjacency vectors, which might pose new problems due to their size and spar- sity. For ML-100K, the lost of some performance when adding content actually contradicts with our initial experiments. In our initial experiments when the RMSE on ML-100K without content was around 0.910, observed that adding content was indeed helpful and reduced the RMSE to
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0.907. However, after introduced ARR and redid the hyperparameter tuning, the RMSE of ML- 100K without content became 0.905, and adding content no longer helped. We hypothesize that the benefits of content reduce with the better modeling of graph structure features.
The way we incorporate content might be another reason for why content not useful in IGMC. In experiments only concatenate the target user and item's 's content vectors with the final graph representation output by the GNN, similar to (Berg et al. (017). However, this method fails model the interactions between content and graph structures in the early graph convolution stage. On the other hand, as Berg et al. (2117) and Zhang & Chen (2018) found, directly concatenating content with initial node features (the one-hot encoding vectors) as the input to GNN often led to worse performance due to information flow bottlenecks. We leave exploring better ways to combine content and graph structures for future work.
14


Reviews:
[{"review_id": "HkevAXKCYB", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper uses a new approach for inductive matrix completion (IMC). The IMC problem is used to model recommender systems where there can be users/items with few (or even no) observed ratings, but where each user/item is associated with additional meta-data or other features. Since users/items close to each other in the feature space plausibly share similar ratings, one can inductively fill in the ratings even in the very sparse regime where traditional matrix completion fails.\n\nThe current paper demonstrates an interesting new approach for IMC that does not require any metadata about new users/items, but rather, exploits the structure of the observed ratings themselves. Specifically, the method (called Inductive Graph-based Matrix Completion, or IGMC) constructs a (fairly large) bipartite graph out of the observed user-item pairs, and creates subgraphs by looking at h-hop neighborhoods of each test node. These subgraphs are fed as features into a graph convolutional neural network (appropriately designed) that predicts any unknown rating.\n\nThe paper is overall well written. The approach (of using subgraph connectivity patterns as features) is clever, novel (at least in my knowledge) and neatly sidesteps the need for extra metadata/features. The numerical results are also thorough and compelling. \n\nMinor comments:\n- Isn't Alg 1 just ordinary bread-first search? Perhaps I missed something.\n- In Table 2 why do some methods have error bars and others don't?\n- Fig 3 is not very informative.\n\n\n", "extra_scores": {"experience_assessment": "I have published in this field for several years.", "review_assessment:_thoroughness_in_paper_reading": "I made a quick assessment of this paper.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory."}, "extra_text": {}}, {"review_id": "H1xK-etpYB", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper presents a method for inductive matrix completion that does not rely on side information to make predictions. The approach is as follows: select the sub-graph from an h-hop neighbourhood of any target user-item pair (u,v) and relabel it according to its distance from (u, v). The authors then treat rating prediction as graph classification problem given the selected subgraph and use a graph convolution architecture to make the prediction. They achieve strong empirical performance in both the transductive and inductive settings. \n\nThe paper presents an interesting approach to the inductive learning problem, with a number of interesting ideas and impressive empirical results. However, I have voted to reject this paper because I found the experimental section did a poor job of showing how the various ideas contribute to overall performance which made it difficult to evaluate the method beyond the absolute performance on benchmark datasets. \n\nThe key idea of the paper is interesting - samples h-hop enclosing subgraphs and label the resulting nodes by their hop-length and type (i.e. users at hop length h get a different label to items at length h).  In their experiments they find that a model with only a 1-hop neighbourhood to performs so well, which suggests that for recommender systems, supporting long chains of dependencies is unnecessary. This is a useful property of the data that the method depends on, because without it, h-hop neighbourhoods with larger h would quickly include the whole graph on most datasets (most real-world graphs have small diameter).\n\nThey process these subgraphs with relational graph convolutions, but make two modifications: first they build their final representations by concatenating all the message passing steps (eq. 2) and both user and item representations (eq. 3), and second they have adjacent rating regularization which constrains weights between adjacent ratings. Eq.2 in particular is unusual because it  amounts to a skip connection from every layer (somewhat analogous to a DenseNet [Huang et al. 2015]). The authors say both lead to improved performance - but don’t evaluate the approaches in the experimental section. What is the contribution of eq 2, 3 and 6 relative to more standard approaches (i.e. no skip connections, sum pooling, no regularization)?\n\nFinally - I found the positioning of the work of the work misleading: from the abstract and related worked we're told that this is a \"seemingly impossible problem\" and prior work has relied on side information to deal with the inductive case. From this description, we might infer that this is the first approach that deals with the no-side information case. But the clear baseline is Hartford et al. 2018 which also doesn't use side information for inductive matrix completion; and yet it is not mentioned in either the introduction or the related work sections. This omission is unfortunate because both because it's misleading, and more importantly it misses the opportunity to discuss the relative merits of the approaches (beyond performance, where this paper provides a clear improvement). For example,\n - How do the two methods compare with respect to computational complexity? This approach seems more expensive because a different subgraph is selected for each prediction (i.e. 1 forward pass per test set point rather than a single forward pass to compute the full test set), but this isn't discussed. \n - Can you give any intuition about where the differences in performance are coming from?\n\n------\n\nIncreased rating to Weak Accept following the rebuttal below.", "extra_scores": {"experience_assessment": "I have published one or two papers in this area.", "review_assessment:_checking_correctness_of_experiments": "I carefully checked the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper thoroughly.", "review_assessment:_checking_correctness_of_derivations_and_theory": "N/A"}, "extra_text": {}}, {"review_id": "BklY_ODfqS", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper presents an inductive matrix completion model using graph neural networks. The proposed method assumes the rating between a user-item pair is determined by this pair's surrounding sub-graph via a neural network. It claims to be an inductive model and don't need any side information as it only uses the surrounding sub-graph structure to give predictions. The experiments also show the promising results on matrix completion tasks, sparse rating matrix cases and transfer learning tasks\n\nThe paper is well written and easy to follow. I think the \"cold-start\" setting this method is mainly focusing on is worth exploring. In this setting, new users have given a few ratings but the quality of their side information is still low. It seems to me that this setting is common in real case. For example, newly registered Netflix users may quickly watch some videos without completing their profile information. However, when the \"new\" users have given some ratings, we can re-train a traditional MC model including the new ratings. But I believe the proposed method of this paper will be useful when re-training cannot be done very frequently. \n\nI have the following concerns or questions.\n\n1. The paper claims the proposed method doesn't need any side information. However, it seems that the side information can be integrated into the model such as concatenating with the one-hot encoding. Have you tried to use the side information for IGMC?\n\n2. The claims in Section 4 don't make much sense to me. For example, it doesn't seem to me that GC-MC can't distinguish Figure (a) and (b) because GC-MC will take the whole graph into consideration. Also, why will node-based approaches push all nodes to have similar embeddings with multiple layers? Could you give some experimental verifications? \n\n3. It's unclear to me how Eq. (6) helps the performance. It will be better to compare a version without using L_{ARR}. Moreover, I believe Eq. (6) can be designed better. \n\n4. GC-MC uses bilinear decoder to give probability predictions for different ratings. Is there any particular reason why you choose squared loss as in Eq. (5) instead of a bilinear decoder? \n\n5. I have a big concern for the scalability. If we want to make recommendations for a user from a million movies, the proposed method seems to need to compute Eq. (4) for one million times. Is there any approximate way to speedup it? \n\nOverall, I think the proposed method is interesting and the experimental results are impressive. \n", "extra_scores": {"experience_assessment": "I have published one or two papers in this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "N/A"}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper proposes a novel technique for matrix completion, using graphical neighborhood structure to side-step the need for any side-information.

Post-rebuttal, the reviewers converged on a unanimous decision to accept. The authors are encouraged to review to address reviewer comments.
Decision: Accept (Spotlight)