Monotone and Separable Set Functions: Characterizations and Neural Models

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 \emph{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 which are monotone by construction, and can approximate all monotone set functions. Experimentally, we consider a variety of set containtment tasks. The experiments show the benefit of using our \our\ model, in comparsion with standard set models which do not incorporate set containment as an inductive bias.