operator distance modulo permutations

The operator distance modulo permutations for an operator $\psi :x\mapsto y$ is given by

This can be defined for any norm on the LHS, but typically is the operator norm induced by the 2 norm.

• We can think of this as a relative measure of how "far" ${\psi }^{\prime }$ is from being a permutation of $\psi$.

For graphs, we can consider $\psi$ to be some graph and $x$ the nodes. This looks for the permutation that makes these graphs closest to one another, then computes a distance.

Let $G$ be a group and $F:x\mapsto y$ an operator. Then the operator distance modulo $G$ is given by

(where, presumably, $F^{\prime }=g_0\circ F$ for some fixed $g_0\in G$)