Min/max stability and box distributions

In representation learning, capturing correlations between the represented elements is paramount. A recent line of work introduces the notion of learning region-based representations, with the objective of being able to better capture these correlations as set interactions. Box models use regions which are products of intervals on $[0,1]$ (i.e., "boxes"), representing joint probability distributions via Lebesgue measure. To mitigate issues with training, a recent work models the endpoints of these intervals using Gumbel distributions, chosen due to their min/max-stability. In this work we analyze min/max-stability on a bounded domain and provide a specific family of such distributions which, replacing Gumbel, allow for stochastic boxes embedded in a finite measure space. This allows for a latent noise model which is a probability measure. Furthermore, we demonstrate an equivalence between this region-based representation and a density representation, where intersection is given by products of densities. We compare our model to previous region-based probability models, and demonstrate it is capable of being trained effectively to modeling correlations.