graph convolutions are stable to perturbations in the data and coefficients

[[concept]]

Takeaway

graph convolutions are Lipschitz stable to perturbations in $x$ and $h_k$.

This is because the convolutional graph filter $y={\sum }_{k=0}^{K-1}{h}_k{S}^{k}x$ is linear in both $h_k$ and $x$, and continuous linear functions are lipschitz continuous.

The convolutional graph filter $y={\sum }_{k=0}^{K-1}{h}_k{S}^{k}x$ is linear in both $h_k$ and $x$.

• (1) if $x^{\prime }=x+ϵ$, then $||y-y^{\prime }||\le (K.{\max }_k{h}_k{\lambda }_i^{++})$
• (2) if $h_k^{\prime }=h_k+ϵ$, then $||y-y^{\prime }||\le ({\lambda }_1^k\cdot ||x||)ϵ$

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