Definition
LightGCN (Light Graph Convolutional Network) is a simplified version of NGCF.
Architecture
Light Graph Convolution
In the LGC, only the normalized sum of neighbor embeddings is performed towards next layer; other operations like self-connection, feature transformation, and nonlinear activation are all removed.
The graph convolution operation in LightGCN is defined as
\mathbf{e}_{u}^{(l+1)} &= \sum_{i \in \mathcal{N}_{u}} \frac{1}{\sqrt{|\mathcal{N}_{u}||\mathcal{N}_{i}|}} \mathbf{e}_{i}^{(l)}\\ \mathbf{e}_{i}^{(l+1)} &= \sum_{u \in \mathcal{N}_{i}} \frac{1}{\sqrt{|\mathcal{N}_{u}||\mathcal{N}_{i}|}} \mathbf{e}_{u}^{(l)} \end{aligned}$$ where - $\mathbf{e}_{u}^{(l)}$ is the embedding of user $u$ at the $l$-th layer - $\mathbf{e}_{i}^{(l)}$ is the embedding of item $i$ at the $l$-th layer - $\mathcal{N}_{u}$ is the set of items interacted with by user $u$. It can be written in a matrix form. $$E^{(l+1)} = \mathcal{L} E^{(l)}$$ where: - $\mathcal{L} = D^{-\frac{1}{2}}AD^{-\frac{1}{2}}$ is the normalized adjacency matrix for the user-item graph - $A = \begin{bmatrix} O&R\\R^{\intercal}&O \end{bmatrix}$ is the [[Adjacency Matrix]], where $R$ is $N \times M$ user-item interaction matrix - $D$ is the [[Degree Matrix]] The only learnable parameters are the embeddings at the $0$-th layer. ## Model Prediction The final user and item embedding matrices are constructed by the weighted sum of the embeddings in each layer. $$E_{\text{final}} = \alpha_{0}E^{(0)} + \alpha_{1}E^{(1)} + \dots + \alpha_{K}E^{(K)}$$ where $\alpha_{k}$ is a hyperparameter (The parameters are set uniformly $\alpha_{k} = \cfrac{1}{K+1}$ in the paper). The score between $u$ and $i$ is calculated using the inner product $$\operatorname{score}_{\text{LGC}}(u, i) = {\mathbf{e}_{u}^{*}}^{\intercal}\mathbf{e}_{i}^{*}$$