Monotone and Separable Set Functions: Characterizations and Neural Models

2025-10-24

AI Summary

This paper addresses the problem of designing vector-valued set-to-vector mapping functions that *exactly* preserve set inclusion: $S subseteq T iff F(S) leq F(T)$ (coordinate-wise). To this end, it introduces the notion of *Monotone and Separating* (MAS) functions, establishes necessary and sufficient conditions for their existence, and derives tight lower bounds on the minimal embedding dimension. For infinite ground sets, it constructs novel models satisfying a relaxed MAS property together with Hölder continuity. Methodologically, the approach unifies set function theory with neural network design, yielding a universal approximator architecture that inherently enforces monotonicity. Experiments demonstrate that the proposed model significantly outperforms standard set encoders—lacking explicit inclusion priors—across multiple set containment prediction tasks, thereby validating the effectiveness and generalization advantage of the introduced inductive bias.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Other Foundations of Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 $Ssubseteq T ext{ 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.

Problem

• Designing set-to-vector functions preserving set containment order
• Establishing dimension bounds for monotone and separable functions
• Creating universal models approximating all monotone set functions

Innovation

• Monotone and Separating set functions preserve order
• Weakly MAS model ensures Holder continuity stability
• Universal models approximate all monotone set functions