Motivated by applications for set containment problems, we consider the following fundamental problem: can we design set-to-vector functions so that the natural partial order on sets is preserved, namely $S\subseteq T \text{ if and only if } F(S)\leq F(T) $. We call functions satisfying this property Monotone and Separating (MAS) set functions. % We establish lower and upper bounds for the vector dimension necessary to obtain MAS functions, as a function of the cardinality of the multisets and the underlying ground set. In the important case of an infinite ground set, we show that MAS functions do not exist, but provide a model called our which provably enjoys a relaxed MAS property we name "weakly MAS" and is stable in the sense of Holder continuity. We also show that MAS functions can be used to construct universal models that are monotone by construction and can approximate all monotone set functions. Experimentally, we consider a variety of set containment tasks. The experiments show the benefit of using our our model, in comparison with standard set models which do not incorporate set containment as an inductive bias. Our code is available in https://github.com/yonatansverdlov/Monotone-Embedding.

翻译：受集合包含问题应用的启发，我们考虑以下基本问题：能否设计从集合到向量的函数，使得集合间的自然偏序关系得以保持，即满足 $S\subseteq T \text{ 当且仅当 } F(S)\leq F(T)$。我们将满足此性质的函数称为单调可分离（MAS）集函数。% 我们针对实现MAS函数所需的向量维度建立了上下界，该维度取决于多重集的基数与基础集的规模。在基础集为无限集的重要情形中，我们证明MAS函数不存在，但提出了一种称为our的模型，该模型可证明地满足我们称为“弱MAS”的松弛性质，并具有赫尔德连续意义下的稳定性。我们还证明了MAS函数可用于构建具有构造单调性的通用模型，该模型能够逼近所有单调集函数。在实验部分，我们考察了多种集合包含任务。实验结果表明，相较于未将集合包含作为归纳偏置的标准集合模型，采用我们的our模型具有显著优势。代码已发布于 https://github.com/yonatansverdlov/Monotone-Embedding。