When “exact recovery” is exact recovery in compressed sensing simulation

In a simulation of compressed sensing (CS), one must test whether the recovered solution x̂ is the true solution x, 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 ϵ
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 (x̂, x), when does “exact recovery” occur with respect to only one or both of these criteria for a given distribution of x? We show that, in a best case scenario, ϵ
²
sets a maximum allowed missed detection rate in a majority sense.