We study the problem of learning permutation invariant representations that can capture containment relations. We propose training a model on a novel task: predicting the size of the symmetric difference between pairs of multisets, sets which may contain multiple copies of the same object. With motivation from fuzzy set theory, we formulate both multiset representations and how to predict symmetric difference sizes given these representations. We model multiset elements as vectors on the standard simplex and multisets as the summations of such vectors, and we predict symmetric difference as the l1-distance between multiset representations. We demonstrate that our representations more effectively predict the sizes of symmetric differences than DeepSets-based approaches with unconstrained object representations. Furthermore, we demonstrate that the model learns meaningful representations, mapping objects of different classes to different standard basis vectors.
Based on fuzzy set theory, we propose a model that given only the sizes of symmetric differences between pairs of multisets, learns representations of such multisets and their elements. This paper proposes a new task of set learning, predicting the size of the symmetric difference between multisets, and gives a method to solve the task based on fuzzy set theory.