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

TL;DR

The paper addresses computing the capacity-distortion-cost function $C(D,B)$ for continuous memoryless channels, where the input cost and state-estimation distortion constrain the achievable rate $I(X;Y)$. It develops a Wasserstein proximal point method in the space of input distributions $\mathcal{P}_2(\mathcal{X})$ and couples it with parametric state estimators $h_\theta$, updated alternately, while using Importance Sampling to approximate intractable integrals. The method is instantiated in an ISAC setting, demonstrating the ability to recover optimal or near-optimal input distributions and estimators across distortion levels and revealing the random-deterministic trade-off as $\beta$ grows. Practically, this enables design insights for waveform and estimator selection in continuous, state-dependent channels and provides a scalable numerical tool beyond discrete finite alphabets.

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.

Figures (3)

• Figure 1: Computed cdc curves for different input power $P$.
• Figure 2: Constellation of particles in 'PM' case with $P=10$dB.
• Figure 3: Constellation of particles in 'NN' case with $P=10$dB.