Since I(w, i) is an indicator function of whether the word w appeared in sentence i, it stands to reason that the product I(w s , i) • I(w t , i) is an indicator of whether both w s and w t appeared in i. Ergo, the numerator of Equation 7is exactly the number of aligned sentences in which both w s and w t occurred: S(w s , w t ).
: S(ws, * )•S( * ,wt) = Dice(ws,wt) 2 8This theoretical result implies that the crosslingual similarity function derived from embeddings based on sentence IDs is essentially a generalization of the Dice coefficient.