Suppose I run code that estimates a set of complex eigenvalues for a matrix operator. Now suppose I re-run this code for $n$ trials, and because of noise, I estimate a new set of eigenvalues every trial.
What's the best predictor of the true spectrum? Is taking an arithmetic mean of the results too naive?
One issue is that we don't know whether the eigenvalues are properly ordered across trials, such that the $i$th eigenvalue for each trial corresponds to the same truth eigenvalue for all $i$. For now, I'm using a lexicographic sort to arrange the eigenvalues in each trial: first sorting by real part and then by imaginary.
One can sort eigenvalues by the complex magnitude (or modulus) in each trial, then compute average value.