Yes, IoU loss is submodular - as a function of the mispredictions

This note is a response to [7] in which it is claimed that [13, Proposition 11] is false. We demonstrate here that this assertion in [7] is false, and is based on a misreading of the notion of set membership in [13, Proposition 11]. We maintain that [13, Proposition 11] is true. Based on the empirical risk principle, one should minimize at training time the loss that one wishes to evaluate at test time [10]. In [12, 13], we have studied the construction of surrogates for loss functions that are submodular with respect to the set of mispredictions.