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Calculating the capacity (with or without feedback) of channels with memoryand continuous alphabets is a challenging task. It requires optimizing thedirected information (DI) rate over all channel input distributions. Theobjective is a multi-letter expression, whose analytic solution is only knownfor a few specific cases. When no analytic solution is present or the channelmodel is unknown, there is no unified framework for calculating or evenapproximating capacity. This work proposes a novel capacity estimationalgorithm that treats the channel as a `black-box', both when feedback is or isnot present. The algorithm has two main ingredients: (i) a neural distributiontransformer (NDT) model that shapes a noise variable into the channel inputdistribution, which we are able to sample, and (ii) the DI neural estimator(DINE) that estimates the communication rate of the current NDT model. Thesemodels are trained by an alternating maximization procedure to both estimatethe channel capacity and obtain an NDT for the optimal input distribution. Themethod is demonstrated on the moving average additive Gaussian noise channel,where it is shown that both the capacity and feedback capacity are estimatedwithout knowledge of the channel transition kernel. The proposed estimationframework opens the door to a myriad of capacity approximation results forcontinuous alphabet channels that were inaccessible until now.

Capacity of Continuous Channels with Memory via Directed Information Neural Estimator