# Improving Set Function Approximation with Quasi-Arithmetic Neural Networks

arXiv: https://arxiv.org/abs/2602.04941 · HTML: https://mlsanity.com/paper/2602.04941/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 31 of its 44 references that are in the map.
- First posted on arXiv 4 Feb 2026.

## Builds on (most cited first, 30 of 31)

- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space](https://mlsanity.com/paper/1706.02413/) (2017)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [ShapeNet: An Information-Rich 3D Model Repository](https://mlsanity.com/paper/1512.03012/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Object-Centric Learning with Slot Attention](https://mlsanity.com/paper/2006.15055/) (2020)
- [Value-Decomposition Networks For Cooperative Multi-Agent Learning](https://mlsanity.com/paper/1706.05296/) (2017)
- [Principal Neighbourhood Aggregation for Graph Nets](https://mlsanity.com/paper/2004.05718/) (2020)
- [Understanding Dimensional Collapse in Contrastive Self-supervised Learning](https://mlsanity.com/paper/2110.09348/) (2021)
- [Bridging the Gap to Real-World Object-Centric Learning](https://mlsanity.com/paper/2209.14860/) (2022)
- [The Reversible Residual Network: Backpropagation Without Storing Activations](https://mlsanity.com/paper/1707.04585/) (2017)
- [Attend, Infer, Repeat: Fast Scene Understanding with Generative Models](https://mlsanity.com/paper/1603.08575/) (2016)
- [SlotFormer: Unsupervised Visual Dynamics Simulation with Object-Centric Models](https://mlsanity.com/paper/2210.05861/) (2022)
- [Sequential Attend, Infer, Repeat: Generative Modelling of Moving Objects](https://mlsanity.com/paper/1806.01794/) (2018)
- [Janossy Pooling: Learning Deep Permutation-Invariant Functions for Variable-Size Inputs](https://mlsanity.com/paper/1811.01900/) (2018)
- [On the Limitations of Representing Functions on Sets](https://mlsanity.com/paper/1901.09006/) (2019)
- [Universal Approximation of Functions on Sets](https://mlsanity.com/paper/2107.01959/) (2021)
- [Improving Object-centric Learning with Query Optimization](https://mlsanity.com/paper/2210.08990/) (2022)
- [FSPool: Learning Set Representations with Featurewise Sort Pooling](https://mlsanity.com/paper/1906.02795/) (2019)
- [On Universal Equivariant Set Networks](https://mlsanity.com/paper/1910.02421/) (2019)
- [Transformers Generalize DeepSets and Can be Extended to Graphs and Hypergraphs](https://mlsanity.com/paper/2110.14416/) (2021)
- [Rep the Set: Neural Networks for Learning Set Representations](https://mlsanity.com/paper/1904.01962/) (2019)
- [Set Norm and Equivariant Skip Connections: Putting the Deep in Deep Sets](https://mlsanity.com/paper/2206.11925/) (2022)
- [Unlocking Slot Attention by Changing Optimal Transport Costs](https://mlsanity.com/paper/2301.13197/) (2023)
- [Learning Representations of Sets through Optimized Permutations](https://mlsanity.com/paper/1812.03928/) (2018)
- [On Deep Set Learning and the Choice of Aggregations](https://mlsanity.com/paper/1903.07348/) (2019)
- [Learning Aggregation Functions](https://mlsanity.com/paper/2012.08482/) (2020)
- [Learnable Commutative Monoids for Graph Neural Networks](https://mlsanity.com/paper/2212.08541/) (2022)

## Abstract (arXiv)

Sets represent a fundamental abstraction across many types of data. To handle the unordered nature of set-structured data, models such as DeepSets and PointNet rely on fixed, non-learnable pooling operations (e.g., sum or max) -- a design choice that can hinder the transferability of learned embeddings and limits model expressivity. More recently, learnable aggregation functions have been proposed as more expressive alternatives. In this work, we advance this line of research by introducing the Neuralized Kolmogorov Mean (NKM) -- a novel, trainable framework for learning a generalized measure of central tendency through an invertible neural function. We further propose quasi-arithmetic neural networks (QUANNs), which incorporate the NKM as a learnable aggregation function. We provide a theoretical analysis showing that, QUANNs are universal approximators for a broad class of common set-function decompositions and, thanks to their invertible neural components, learn more structured latent representations. Empirically, QUANNs outperform state-of-the-art baselines across diverse benchmarks, while learning embeddings that transfer effectively even to tasks that do not involve sets.