Graph Neural Networks

Data

subject:: Data Science Methods for Large Scale Graphs parent:: Graph Signals and Graph Signal Processing theme:: math notes

GNN

A “full-fledged” GNN layer is a multi-dimensional generalization of the multi-layer graph perceptron given by $x_{\ell }=\sigma (U_{\ell })=\sigma \left({\sum }_{k=0}^{K-1}{S}^k{x}_{\ell -1}{H}_{\ell ,k}\right)$ $H_{\ell ,k}\in {\mathbb{R}}^{d_{\ell -1}\times d_{\ell }}$ for $1\le \ell \le L$, where

• $X_0=X\in {\mathbb{R}}^{n\times d_0}$
• $Y=X_L\in {\mathbb{R}}^{n\times d_{\ell }}$
• $\sigma :\mathbb{R}\to \mathbb{R}$ is a pointwise nonlinearity
• and the multiplication is a convolutional filter bank

Here, $X_{\ell }\in {\mathbb{R}}^{n\times d_{\ell }}$ is still called an $\ell$-layer embedding.

Note

These GNNs are sometimes called convolutional GNNs because they are based on graph convolution.

GNN layer equation

$X_{\ell }=\sigma (U_{\ell })=\sigma \left({\sum }_{k=0}^{K-1}{S}^k{X}_{\ell -1}{H}_{\ell ,k}\right)$

Mentions

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