When 'exact recovery' is exact recovery in compressed sensing simulation

In a simulation of compressed sensing (CS), one must test whether the recovered solution \(\vax\) is the true solution \(\vx\), i.e., ``exact recovery.''
Most CS simulations employ one of two criteria: 1) the recovered support is the true support; or 2) the normalized squared error is less than \(\epsilon^2\). We analyze these exact recovery criteria independent of any recovery algorithm, but with respect to signal distributions that are often used in CS simulations. That is, given a pair \((\vax,\vx)\), when does ``exact recovery'' occur with respect to only one or both of these criteria for a given distribution of \(\vx\)? We show that, in a best case scenario, \(\epsilon^2\) sets a maximum allowed missed detection rate
in a majority sense.