filter permutation invariance

[[concept]]

invariant to permutations

Let $S$ be a graph shift operator. A graph filter $H(S)$ is permutation invariant if $S$ is permutation invariant.

If $S$ is permutation invariant, then since $H$ is also permutation invariant, we have $||S-S^{\prime }|{|}_p=0⟹||H(S)-H(S^{\prime })|{|}_p=0$ Where $||\cdot |{|}_p$ is the operator distance modulo permutations

For GNNs, we have equivalently $||S-S^{\prime }|{|}_p=0⟹||\Phi (S)-\Phi (S^{\prime })|{|}_p=0$ since this is a composed graph convolution

permutation invariant (alternate definition)

Recall the definition for invariant:

Invariant $G$ be a group acting on a (data) set $X$. Then $F:X\to Y$ is invariant if $F(g \circ x) = F(x) ;;; \forall ; g \in G \text{ and } x \in X$$

Let

If $G$ is a permutation group and $F$, then for any $F^{\prime }=g\circ F$, we have $F$ is permutation invariant if and only if $||F-F^{\prime }|{|}_{p(=g)}=0$ That is, $F$ is invariant to permutations if and only if its operator distance modulo permutations is 0.

This follows directly from the definition of invariance.

Mentions

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