# Joint Learning of Geometric and Probabilistic Constellation Shaping

arXiv: https://arxiv.org/abs/1906.07748 · HTML: https://mlsanity.com/paper/1906.07748/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Jun 2019.

## Built on by (most cited first, 13 of 13)

- [DeepJSCC-Q: Channel Input Constrained Deep Joint Source-Channel Coding](https://mlsanity.com/paper/2111.13042/) (2021)
- [Bit Error and Block Error Rate Training for ML-Assisted Communication](https://mlsanity.com/paper/2210.14103/) (2022)
- [Diffusion Models for Wireless Communications](https://mlsanity.com/paper/2310.07312/) (2023)
- [Probabilistic Constellation Shaping With Denoising Diffusion Probabilistic Models: A Novel Approach](https://mlsanity.com/paper/2309.08688/) (2023)
- [End-to-End Deep Learning of Long-Haul Coherent Optical Fiber Communications via Regular Perturbation Model](https://mlsanity.com/paper/2107.12320/) (2021)
- [End-to-End Learning of Joint Geometric and Probabilistic Constellation Shaping](https://mlsanity.com/paper/2112.05050/) (2021)
- [Hybrid Neural Coded Modulation: Design and Training Methods](https://mlsanity.com/paper/2202.01972/) (2022)
- [Model-Based Deep Learning of Joint Probabilistic and Geometric Shaping for Optical Communication](https://mlsanity.com/paper/2204.07457/) (2022)
- [Generative AI-Based Probabilistic Constellation Shaping With Diffusion Models](https://mlsanity.com/paper/2311.09349/) (2023)
- [Discriminative Mutual Information Estimators for Channel Capacity Learning](https://mlsanity.com/paper/2107.03084/) (2021)
- [Neural Probabilistic Shaping: Joint Distribution Learning for Optical Fiber Communications](https://mlsanity.com/paper/2507.16012/) (2025)
- [Not All Symbols Are Equal: Importance-Aware Constellation Design for Semantic Communication](https://mlsanity.com/paper/2605.14940/) (2026)
- [A Perspective on Neural Capacity Estimation: Viability and Reliability](https://mlsanity.com/paper/2203.11793/) (2022)

## Abstract (arXiv)

The choice of constellations largely affects the performance of communication systems. When designing constellations, both the locations and probability of occurrence of the points can be optimized. These approaches are referred to as geometric and probabilistic shaping, respectively. Usually, the geometry of the constellation is fixed, e.g., quadrature amplitude modulation (QAM) is used. In such cases, the achievable information rate can still be improved by probabilistic shaping. In this work, we show how autoencoders can be leveraged to perform probabilistic shaping of constellations. We devise an information-theoretical description of autoencoders, which allows learning of capacity-achieving symbol distributions and constellations. Recently, machine learning techniques to perform geometric shaping were proposed. However, probabilistic shaping is more challenging as it requires the optimization of discrete distributions. Furthermore, the proposed method enables joint probabilistic and geometric shaping of constellations over any channel model. Simulation results show that the learned constellations achieve information rates very close to capacity on an additive white Gaussian noise (AWGN) channel and outperform existing approaches on both AWGN and fading channels.