spectral.tex (942B)
1 \begin{frame}{GCN Spatial \& Spectral}% 2 \begin{block}{Spectral (convolution is multiplication in Fourier space)} 3 \centering% 4 \begin{tabular}{l l l} 5 & \strong{Graph} & \strong{Euclidean} \\ 6 \midrule 7 Laplacian & \(\mtrx{L}=\mtrx{D}-\mtrx{M}\) & \(\nabla^2\) \\ 8 \(\hookrightarrow\) Eigenfunctions & \(\mtrx{U}\) s.t.~\(\mtrx{L}=\mtrx{U}\mtrx{\Lambda}\mtrx{U}^{-1}\) & \(\xi\mapsto e^{2\pi i\xi x}\) \\ 9 Fourier transform & \(\mtrx{U}\transpose\vctr{f}\) & \(\gffourier(f) = \int_{-\infty}^{\infty} f(x) e^{2\pi i\xi x} \diff x\) \\ 10 Convolution & \(\mtrx{U}(\mtrx{U}\transpose\vctr{w}\mtrx{U}\transpose\vctr{f})\) & \(\gfinvfourier(\gffourier(w)\gffourier(f))\) 11 \end{tabular} 12 \end{block} 13 \begin{block}{Spatial} 14 \centering% 15 \(\displaystyle\operatorname{\text{GCN}}(\mtrx{X}; \mtrx{W})_v = \ReLU\left(\frac{1}{|\gfneighbors(v)|} \sum_{n_i\in\gfneighbors(v)} \mtrx{W} \mtrx{X}_{n_i} \right)\) 16 \end{block} 17 \end{frame}