# Fixed-Dimensional and Permutation Invariant State Representation of Autonomous Driving

arXiv: https://arxiv.org/abs/2105.11299 · HTML: https://mlsanity.com/paper/2105.11299/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 May 2021.

## Built on by (most cited first, 5 of 5)

- [Encoding Distributional Soft Actor-Critic for Autonomous Driving in Multi-lane Scenarios](https://mlsanity.com/paper/2109.05540/) (2021)
- [Self-learned Intelligence for Integrated Decision and Control of Automated Vehicles at Signalized Intersections](https://mlsanity.com/paper/2110.12359/) (2021)
- [Risk-Aware Reinforcement Learning for Autonomous Driving: Improving Safety When Driving through Intersection](https://mlsanity.com/paper/2503.19690/) (2025)
- [A Survey of Reinforcement Learning-Based Motion Planning for Autonomous Driving: Lessons Learned from a Driving Task Perspective](https://mlsanity.com/paper/2503.23650/) (2025)
- [Integrated Decision and Control for High-Level Automated Vehicles by Mixed Policy Gradient and Its Experiment Verification](https://mlsanity.com/paper/2210.10613/) (2022)

## Abstract (arXiv)

In this paper, we propose a new state representation method, called encoding sum and concatenation (ESC), for the state representation of decision-making in autonomous driving. Unlike existing state representation methods, ESC is applicable to a variable number of surrounding vehicles and eliminates the need for manually pre-designed sorting rules, leading to higher representation ability and generality. The proposed ESC method introduces a representation neural network (NN) to encode each surrounding vehicle into an encoding vector, and then adds these vectors to obtain the representation vector of the set of surrounding vehicles. By concatenating the set representation with other variables, such as indicators of the ego vehicle and road, we realize the fixed-dimensional and permutation invariant state representation. This paper has further proved that the proposed ESC method can realize the injective representation if the output dimension of the representation NN is greater than the number of variables of all surrounding vehicles. This means that by taking the ESC representation as policy inputs, we can find the nearly optimal representation NN and policy NN by simultaneously optimizing them using gradient-based updating. Experiments demonstrate that compared with the fixed-permutation representation method, the proposed method improves the representation ability of the surrounding vehicles, and the corresponding approximation error is reduced by 62.2%.