# Permutation-Invariant Set Autoencoders with Fixed-Size Embeddings for Multi-Agent Learning

arXiv: https://arxiv.org/abs/2302.12826 · HTML: https://mlsanity.com/paper/2302.12826/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 24 references that are in the map.
- First posted on arXiv 24 Feb 2023.

## Builds on (most cited first, 21 of 21)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Non-local Neural Networks](https://mlsanity.com/paper/1711.07971/) (2017)
- [Modeling Relational Data with Graph Convolutional Networks](https://mlsanity.com/paper/1703.06103/) (2017)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [On the Properties of Neural Machine Translation: Encoder-Decoder Approaches](https://mlsanity.com/paper/1409.1259/) (2014)
- [QMIX: Monotonic Value Function Factorisation for Deep Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/1803.11485/) (2018)
- [Linear Transformers Are Secretly Fast Weight Programmers](https://mlsanity.com/paper/2102.11174/) (2021)
- [Stochastic Latent Actor-Critic: Deep Reinforcement Learning with a Latent Variable Model](https://mlsanity.com/paper/1907.00953/) (2019)
- [Going Beyond Linear Transformers with Recurrent Fast Weight Programmers](https://mlsanity.com/paper/2106.06295/) (2021)
- [Graph Neural Networks for Decentralized Multi-Robot Path Planning](https://mlsanity.com/paper/1912.06095/) (2019)
- [Deep Set Prediction Networks](https://mlsanity.com/paper/1906.06565/) (2019)
- [Graph Convolutional Encoders for Syntax-aware Neural Machine Translation](https://mlsanity.com/paper/1704.04675/) (2017)
- [FSPool: Learning Set Representations with Featurewise Sort Pooling](https://mlsanity.com/paper/1906.02795/) (2019)
- [Conditional Set Generation with Transformers](https://mlsanity.com/paper/2006.16841/) (2020)
- [QGNN: Value Function Factorisation with Graph Neural Networks](https://mlsanity.com/paper/2205.13005/) (2022)
- [3DCNN-DQN-RNN: A Deep Reinforcement Learning Framework for Semantic Parsing of Large-scale 3D Point Clouds](https://mlsanity.com/paper/1707.06783/) (2017)
- [ModGNN: Expert Policy Approximation in Multi-Agent Systems with a Modular Graph Neural Network Architecture](https://mlsanity.com/paper/2103.13446/) (2021)
- [Hierarchical Deep Multiagent Reinforcement Learning with Temporal Abstraction](https://mlsanity.com/paper/1809.09332/) (2018)

## Built on by (most cited first, 4 of 4)

- [RAPiD-Seg: Range-Aware Pointwise Distance Distribution Networks for 3D LiDAR Segmentation](https://mlsanity.com/paper/2407.10159/) (2024)
- [Generalising Multi-Agent Cooperation through Task-Agnostic Communication](https://mlsanity.com/paper/2403.06750/) (2024)
- [Learning to Sample: Reinforcement Learning-Guided Sampling for Autonomous Vehicle Motion Planning](https://mlsanity.com/paper/2509.24313/) (2025)
- [GLASS: Global Latent Aggregation with Slot-based Set Decoding for Scalable All-Atom Crystal Generation](https://mlsanity.com/paper/2609.37158/) (2026)

## Abstract (arXiv)

The problem of permutation-invariant learning over set representations is particularly relevant in the field of multi-agent systems -- a few potential applications include unsupervised training of aggregation functions in graph neural networks (GNNs), neural cellular automata on graphs, and prediction of scenes with multiple objects. Yet existing approaches to set encoding and decoding tasks present a host of issues, including non-permutation-invariance, fixed-length outputs, reliance on iterative methods, non-deterministic outputs, computationally expensive loss functions, and poor reconstruction accuracy. In this paper we introduce a Permutation-Invariant Set Autoencoder (PISA), which tackles these problems and produces encodings with significantly lower reconstruction error than existing baselines. PISA also provides other desirable properties, including a similarity-preserving latent space, and the ability to insert or remove elements from the encoding. After evaluating PISA against baseline methods, we demonstrate its usefulness in a multi-agent application. Using PISA as a subcomponent, we introduce a novel GNN architecture which serves as a generalised communication scheme, allowing agents to use communication to gain full observability of a system.