LightGCN

Summarizing of the paper

Head - LightGCN: Simplifying and Powering Graph Convolution Network for Recommendation.

Algorithms:

$$\mathbf{e}^{k+1}_u=\sum_{i\in N_u}\dfrac{1}{\sqrt{|N_u|}\sqrt{|N_i|}}\mathbf{e}^{(k)}_i \\ \mathbf{e}^{k+1}_i=\sum_{u\in N_i}\dfrac{1}{\sqrt{|N_i|}\sqrt{|N_u|}}\mathbf{e}^{(k)}_u$$

The final representation is the form of combined layer embeddings.$\mathbf{e}_u=\sum^K_{k=0}\alpha_k \mathbf{e}_u^{(k)}; \;\; \mathbf{e}_i=\sum^K_{k=0}\alpha_k\mathbf{e}^{(k)}_i$

The model prediction is defined as the inner product of user and item final representations: $\hat{y}_{ui}=\mathbf{e}^T_u\mathbf{e}_i$ . It implies the similarity between the user and item.

Matrix Form:

$$\mathbf{A}=\begin{bmatrix} \mathbf{0} \;\;\; \;\mathbf{R} \\ \mathbf{R}^T \;\; \mathbf{0} \end{bmatrix} , \;\; \mathbf{E}^{(k+1)}=(\mathbf{D}^{-1/2}\mathbf{A}\mathbf{D}^{-1/2})\mathbf{E}^{(k)}$$

• $\mathbf{R}$ is a $M \times N$ user-item interaction matrix. Each entries 1 if $u$ is connected to $i$

• $\mathbf{D}$is a $(M+N)\times(M+N)$ diagonal matrix, in which each entry $D_{ii}$ denotes the number of nonzero entries in the $i_{th}$row vector of $\mathbf{A}$

• $\mathbf{E}$ is a $(M+N)\times T$matrix where $T$ is the embedding size.

We easily make this as a code using torch_geomtric.utils (reference)