convolutional graph filters are shift equivariant

Data

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

Theorem

graph convolution are shift equivariant.

Let $y=H(S)x$ where $S$ is a shift operator. Suppose $x^{\prime }=Sx$. To show shift equivariance, we need to show $y^{\prime }=Sy$ (ie the output is shifted in the same way the input was).

Proof

y’ = H(S)x’ &= \sum_{k=0}^{K-1}h_{k}S^{k} (Sx) \ &= \sum_{k=0}^{K-1}h_{k}S^{k+1}x \ &= S \sum_{k=0}^{K-1}h_{k}S^k x \ &= Sy \end{aligned}$$

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