For simplicity, we use the following two abbreviations: C s := C ∪ {s} and  Therefore, we obtain f NN C (s) ≥ 0 from the nondecreasing property of g. (Submodularity) Next, we prove the submodularity.
for any two sets C 1 and C 2 of sentences such that C 1 ⊂ C 2 ⊂ D, and sentence s ∈ D \C 2 , we have we obtain α ≥ 0 from the non-decreasing property of g. If The objective function f NN is simply heuristic for small documents, but the next theorem shows that f NN is asymptotically related to an approximation of KL-divergence in a continuous space, if g is a logarithmic function.