# Learning Representations of Sets through Optimized Permutations

arXiv: https://arxiv.org/abs/1812.03928 · HTML: https://mlsanity.com/paper/1812.03928/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Dec 2018.

## Built on by (most cited first, 15 of 15)

- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [Equivariant Frames and the Impossibility of Continuous Canonicalization](https://mlsanity.com/paper/2402.16077/) (2024)
- [GANzzle: Reframing jigsaw puzzle solving as a retrieval task using a generative mental image](https://mlsanity.com/paper/2207.05634/) (2022)
- [Set-to-Sequence Methods in Machine Learning: a Review](https://mlsanity.com/paper/2103.09656/) (2021)
- [Positional Diffusion: Ordering Unordered Sets with Diffusion Probabilistic Models](https://mlsanity.com/paper/2303.11120/) (2023)
- [Advances in Set Function Learning: A Survey of Techniques and Applications](https://mlsanity.com/paper/2501.14991/) (2025)
- [Adaptive Canonicalization with Application to Invariant Anisotropic Geometric Networks](https://mlsanity.com/paper/2509.24886/) (2025)
- [Set Interdependence Transformer: Set-to-Sequence Neural Networks for Permutation Learning and Structure Prediction](https://mlsanity.com/paper/2206.03720/) (2022)
- [Towards Improving the Generation Quality of Autoregressive Slot VAEs](https://mlsanity.com/paper/2206.01370/) (2022)
- [Set Twister for Single-hop Node Classification](https://mlsanity.com/paper/2112.09752/) (2021)
- [GausSetExpander: A Simple Approach for Entity Set Expansion](https://mlsanity.com/paper/2202.13649/) (2022)
- [Effective and Interpretable Information Aggregation with Capacity Networks](https://mlsanity.com/paper/2207.12013/) (2022)
- [Approximating invariant functions with the sorting trick is theoretically justified](https://mlsanity.com/paper/2403.01671/) (2024)
- [Improving Set Function Approximation with Quasi-Arithmetic Neural Networks](https://mlsanity.com/paper/2602.04941/) (2026)
- [Distributionally Robust Set Representation Learning Under Inference-Time Element Corruption](https://mlsanity.com/paper/2605.30089/) (2026)

## Abstract (arXiv)

Representations of sets are challenging to learn because operations on sets should be permutation-invariant. To this end, we propose a Permutation-Optimisation module that learns how to permute a set end-to-end. The permuted set can be further processed to learn a permutation-invariant representation of that set, avoiding a bottleneck in traditional set models. We demonstrate our model's ability to learn permutations and set representations with either explicit or implicit supervision on four datasets, on which we achieve state-of-the-art results: number sorting, image mosaics, classification from image mosaics, and visual question answering.