设计可保持集合包含关系的神经模型，提升集合推理准确性。
Monotone and Separable Set Functions: Characterizations and Neural Models
- 构建满足单调性与分离性的集合函数，保证子集关系映射一致。
- 在无限基集下证明严格单调函数不存在，提出弱单调稳定模型。
- 实验显示该模型在集合包含任务中优于传统方法，适合结构化推理场景。
针对集合包含问题，本文研究如何设计集合到向量的函数，使集合的自然偏序关系得以保留：当且仅当 $S\⊆ T$ 时，有 $F(S)\leq F(T)$。满足此性质的函数称为单调且分离（MAS）集合函数。我们建立了向量维数的上下界，依赖于多重集大小与基集规模。在无限基集情况下，证明严格MAS函数不存在，但提出名为 our 的模型，其具备可证明的“弱MAS”性质，并在Holder连续意义下稳定。此外，展示MAS函数可用于构造天生单调的通用模型，能逼近所有单调集合函数。实验验证了 our 模型在多种集合包含任务中的优势，相较于未引入包含先验的标准集合模型表现更优。代码已开源：https://github.com/structlearning/MASNET。
原文摘要 · Abstract (English)
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/structlearning/MASNET.
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