# Monotone and Separable Set Functions: Characterizations and Neural Models

arXiv: https://arxiv.org/abs/2510.23634 · HTML: https://mlsanity.com/paper/2510.23634/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 47 references that are in the map.
- First posted on arXiv 24 Oct 2025.

## Builds on (most cited first, 20 of 20)

- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [The Expressive Power of Neural Networks: A View from the Width](https://mlsanity.com/paper/1709.02540/) (2017)
- [Analyzing Inverse Problems with Invertible Neural Networks](https://mlsanity.com/paper/1808.04730/) (2018)
- [Unconstrained Monotonic Neural Networks](https://mlsanity.com/paper/1908.05164/) (2019)
- [On the Limitations of Representing Functions on Sets](https://mlsanity.com/paper/1901.09006/) (2019)
- [Deep Lattice Networks and Partial Monotonic Functions](https://mlsanity.com/paper/1709.06680/) (2017)
- [Neural Injective Functions for Multisets, Measures and Graphs via a Finite Witness Theorem](https://mlsanity.com/paper/2306.06529/) (2023)
- [Permutation Invariant Representations with Applications to Graph Deep Learning](https://mlsanity.com/paper/2203.07546/) (2022)
- [Neural Set Function Extensions: Learning with Discrete Functions in High Dimensions](https://mlsanity.com/paper/2208.04055/) (2022)
- [DESSERT: An Efficient Algorithm for Vector Set Search with Vector Set Queries](https://mlsanity.com/paper/2210.15748/) (2022)
- [FSW-GNN: A Bi-Lipschitz WL-Equivalent Graph Neural Network](https://mlsanity.com/paper/2410.09118/) (2024)
- [Fourier Sliced-Wasserstein Embedding for Multisets and Measures](https://mlsanity.com/paper/2504.02544/) (2025)
- [Mini-Batch Consistent Slot Set Encoder for Scalable Set Encoding](https://mlsanity.com/paper/2103.01615/) (2021)
- [Neural Estimation of Submodular Functions with Applications to Differentiable Subset Selection](https://mlsanity.com/paper/2210.11033/) (2022)
- [Polynomial Width is Sufficient for Set Representation with High-dimensional Features](https://mlsanity.com/paper/2307.04001/) (2023)
- [On the H"older Stability of Multiset and Graph Neural Networks](https://mlsanity.com/paper/2406.06984/) (2024)
- [Scalable Set Encoding with Universal Mini-Batch Consistency and Unbiased Full Set Gradient Approximation](https://mlsanity.com/paper/2208.12401/) (2022)
- [Monotonic Learning in the PAC Framework: A New Perspective](https://mlsanity.com/paper/2501.05493/) (2025)

## Abstract (arXiv)

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  if and only if  F(S)≤ 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.