Sorry about the late answer. This helped me, thanks a lot!

@Stanislav: In my case, A is a function of a a lot of parameters, and takes ~0.1s to calculate (Monte Carlo simulation). I'm trying to optimise the ratio with regard to those parameters. But for this question, I'm just asking if some properties of these ratios have been investigated before.

@Stanislav: It is indeed. The only assumption on $A$ so far is that the expected energy of each sample series is equal, i.e. the 2-norm of each column of $A$ is roughly equal.

@Anthony: The matrix for my application originates from a noisy sampling process, so the denominator is almost surely positive.