I liked this paper's clear introduction to the task at hand.

I believe that the m that people are discussing is simply the sample training size, which can vary according to the experimenter's will. Certainly m < n, but other than that the authors could have limited m so that the graphs showed more detail. Of course, this leaves the reader to wonder what happens if m could get much bigger but they don't show it on the graphs.

They also make a case in lemma 3 and its proof that m = O(log(n)) for a generative model, which is probably another reason to justify limiting m in their experiments.