operator distance modulo permutations

[[concept]]

operator distance modulo permutations

The operator distance modulo permutations for an operator $\psi :x\mapsto y$ is given by $\begin{array}{rl}||\psi -{\psi }^{\prime }|{|}_p& =\underset{P\in \mathcal{P}}{\min }\underset{x:||x||=1}{\max }||P^T\psi (x)-{\psi }^{\prime }(P^{T}x)||\\ & =\underset{x:||x||=1}{\sup }||P^T\psi (x)-{\psi }^{\prime }(P^{T}x)||\end{array}$ 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 $\psi$.

NOTE

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.

operator distance modulo $G$

Let $G$ be a group and $F:x\mapsto y$ an operator. Then the operator distance modulo $G$ is given by $||F-F^{\prime }|{|}_G={\min }_{g\in G}{\max }_{x:||x||=1}||g\circ F(x)-F^{\prime }(g\circ x)||$ (where, presumably, $F^{\prime }=g_0\circ F$ for some fixed $g_0\in G$)

Mentions

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