Monotone and Separable Set Functions: Characterizations and Neural Models

SOUTRIK SARANGI, Yonatan Sverdlov, Nadav Dym, Abir De

Keywords: DeepSets, Monotone functions, Data Retrieval, separability

TL;DR: We build a multi-set function that is monotone and separable

Abstract:

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 $S \subseteq T$ 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 MASNET
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 MASNET model, in comparison with standard set models which do not
incorporate set containment as an inductive bias.

Primary Area: Other (please use sparingly, only use the keyword field for more details)

Submission Number: 10783