To make matrix completion inductive, most previous works use content(side information) to make prediction.
Problems: Highquality content is noe always available and can be hard to extract
Conventional methods: lowrank rating matrix decomposition. In IMC: r_{ij}=x_i^T Q y_j, where Q is a learnable matrix modeling the feature interactionms. IMC methods have strong constraints on the content quality.
In 2018, Hartford propose exchangeable matrix layers which do not rely on side information while needing the whole rating matrix as the input, which tends to lead to a scalability challenges for practical datasets.
In this paper, a novel inductive matrix completion method that does not use any content. The key component is local graph pattern. Predicting the unknown ratings converts equivalentl to predicting labeled links in the corresponding bipartite graphs.
IGMC uses a graphlevel GNN to learn representations for subgraphs.
IGMC
 Enclosing subgraph extraction
BFS procedure for extracting hhop enclosing subgraphs:
 Node labeling
Assign an integer to every node in the subgraph. The purpose is to mark different roles for nodes.
 target user and target item

Usertype nodes and itemtype nodes
0, 1 > 2i, 2i+1
 IGMC purely relies on graph patterns within local enclosing subgraphs. > inductive (compared to GCMC 2018, which is a transductive method)
 Graph neural network architecture
Two key components:
 Message passing layers: extract a feature vector for each node in the subgraph.
RGCN:
$$
\mathbf{x}_{i}^{l+1}=\mathbf{W}_{0}^{l} \mathbf{x}_{i}^{l}+\sum_{r \in \mathcal{R}} \sum_{j \in \mathcal{N}_{r}(i)} \frac{1}{\left\mathcal{N}_{r}(i)\right} \mathbf{W}_{r}^{l} \mathbf{x}_{j}^{l}$$
$$\mathbf{h}_{i}=\operatorname{concat}\left(\mathbf{x}_{i}^{1}, \mathbf{x}_{i}^{2}, \ldots, \mathbf{x}_{i}^{L}\right)$$
 A pooling layer: summarize a subgraph representation from node features
\mathbf{g}=\operatorname{concat}\left(\mathbf{h}_{u}, \mathbf{h}_{v}\right), \quad
\hat{r}=\mathbf{w}^{\top} \sigma(\mathbf{W} \mathbf{g})
 Model training
Loss function:
$$\mathcal{L}=\frac{1}{{(u, v) \mid \mathbf{\Omega}_{u, v}=1}} \sum_{(u, v): \boldsymbol{\Omega}_{u, v}=1}\left(R_{u, v}\hat{R}_{u, v}\right)^{2}$$
$$\mathcal{L}_{\mathrm{ARR}}=\sum_{i=1,2, \ldots,\mathcal{R}1}\left\Vert\mathbf{W}_{r_{i+1}}\mathbf{W}_{r_{i}}\right\Vert_{F}^{2}$$
$$
\mathcal{L}_{\mathrm{final}} = \mathcal{L} + \lambda \mathcal{L}_{\mathrm{ARP}}$$
$\mathcal{L}_{\mathrm{ARR}}$ encourages rating adjacent to each other to have similar parameter matrices.
 Comparison to GCMC
 IGMC uses a graphlevel GNN to map the enclosing subgraph whereas it has a higher complexity.

GCMC uses a nodelevel GNN on the bipartite graph G to learn embeddings to predict the rating. The main drawback is that it fails to model interactions and correspondences between the nodes due to the independent consideration for nodes.
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