  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.
