End-to-End Learning of Probabilistic Constellation Shaping through Importance Sampling

Probabilistic constellation shaping enables easy rate adaption and has been proven to reduce the gap to Shannon capacity. Constellation point probabilities are optimized to maximize either the mutual information or the bit-wise mutual information. The optimization problem is however challenging even for simple channel models. While autoencoder-based machine learning has been applied successfully to solve this problem [1], it requires manual computation of additional terms for the gradient which is an error-prone task. In this work, we present novel loss functions for autoencoder-based learning of probabilistic constellation shaping for coded modulation systems using automatic differentiation and importance sampling. We show analytically that our proposed approach also uses exact gradients of the constellation point probabilities for the optimization. In simulations, our results closely match the results from [1] for the additive white Gaussian noise channel and a simplified model of the intensity-modulation direct-detection channel.