Computation of Capacity-Distortion-Cost Functions for Continuous Memoryless Channels

Abstract: This paper aims at computing the capacity-distortion-cost (CDC) function for continuous memoryless channels, which is defined as the supremum of the mutual information between channel input and output, constrained by an input cost and an expected distortion of estimating channel state. Solving the optimization problem is challenging because the input distribution does not lie in a finite-dimensional Euclidean space and the optimal estimation function has no closed form in general. We propose to adopt the Wasserstein proximal point method and parametric models such as neural networks (NNs) to update the input distribution and estimation function alternately. To implement it in practice, the importance sampling (IS) technique is used to calculate integrals numerically, and the Wasserstein gradient descent is approximated by pushing forward particles. The algorithm is then applied to an integrated sensing and communications (ISAC) system, validating theoretical results at minimum and maximum distortion as well as the random-deterministic trade-off.