Data
subject:: Data Science Methods for Large Scale Graphs parent:: Graph Signals and Graph Signal Processing theme:: math notes
Theorem
graph convolution are local.
steps:
We can write locally in terms of
y &= \sum_{k=0}^{K-1}h_{k}S^k x \ &= \sum_{k=0}^{K-1}h_{k}S^{k-1}Sx, ;; \text{let } z_{1}= Sx\ &= \sum_{k=0}^{K-1}h_{k}S^{k-1}z_{1} + h_{0}x \ &= \sum_{k=1}^{K-1}h_{k} S^{k-2}Sz_{1} + h_{0}x, ;; \text{let } z_{2}=Sz_{1}\ &= \sum_{k=2}^{K-1}h_{k}S^{k-2}Sz_{2} + h_{0}x + h_{1}z_{1} \ &= \dots\ &= \sum_{k=0}^{K-1}h_{k}z_{k} ,, (\text{letting }z_{i}=S^ix) \end{aligned}$$
Visual representation of the stepwise local expression
Mentions
TABLE
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