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Computing Capacity-Cost Functions for Continuous Channels in Wasserstein Space

Xinyang Li, Vlad C. Andrei, Ullrich J. Mönich, Fan Liu, Holger Boche

TL;DR

The paper addresses computing the capacity-cost function for continuous channels by formulating a Wasserstein-proximal gradient flow in the space of probability measures. It replaces the KL proximal penalty with the Wasserstein distance $W_2$, derives transport-based input updates via the Kantorovich potential, and implements a particle-based scheme with importance sampling to handle intractable integrals. The authors provide convergence guarantees under suitable step-size conditions and demonstrate the approach on MIMO-AWGN and fading channels, including CSIR scenarios, with results aligning to theory. Additionally, the framework unifies the computation of the rate-distortion function under the same Wasserstein proximal paradigm, offering a scalable, rigorous method for continuous-space information-theoretic problems.

Abstract

This paper investigates the problem of computing capacity-cost (C-C) functions for continuous channels. Motivated by the Kullback-Leibler divergence (KLD) proximal reformulation of the classical Blahut-Arimoto (BA) algorithm, the Wasserstein distance is introduced to the proximal term for the continuous case, resulting in an iterative algorithm related to the Wasserstein gradient descent. Practical implementation involves moving particles along the negative gradient direction of the objective function's first variation in the Wasserstein space and approximating integrals by the importance sampling (IS) technique. Such formulation is also applied to the rate-distortion (R-D) function for continuous source spaces and thus provides a unified computation framework for both problems.

Computing Capacity-Cost Functions for Continuous Channels in Wasserstein Space

TL;DR

The paper addresses computing the capacity-cost function for continuous channels by formulating a Wasserstein-proximal gradient flow in the space of probability measures. It replaces the KL proximal penalty with the Wasserstein distance , derives transport-based input updates via the Kantorovich potential, and implements a particle-based scheme with importance sampling to handle intractable integrals. The authors provide convergence guarantees under suitable step-size conditions and demonstrate the approach on MIMO-AWGN and fading channels, including CSIR scenarios, with results aligning to theory. Additionally, the framework unifies the computation of the rate-distortion function under the same Wasserstein proximal paradigm, offering a scalable, rigorous method for continuous-space information-theoretic problems.

Abstract

This paper investigates the problem of computing capacity-cost (C-C) functions for continuous channels. Motivated by the Kullback-Leibler divergence (KLD) proximal reformulation of the classical Blahut-Arimoto (BA) algorithm, the Wasserstein distance is introduced to the proximal term for the continuous case, resulting in an iterative algorithm related to the Wasserstein gradient descent. Practical implementation involves moving particles along the negative gradient direction of the objective function's first variation in the Wasserstein space and approximating integrals by the importance sampling (IS) technique. Such formulation is also applied to the rate-distortion (R-D) function for continuous source spaces and thus provides a unified computation framework for both problems.
Paper Structure (12 sections, 2 theorems, 34 equations, 4 figures, 1 algorithm)

This paper contains 12 sections, 2 theorems, 34 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background and Notations
  3. Blahut-Arimoto Algorithm
  4. Wasserstein Space
  5. Proposed Method
  6. Problem Formulation
  7. Alternating Updates with Wasserstein Gradient Descent
  8. Numerical Implementation
  9. Rate-Distortion Function
  10. Convergence Analysis
  11. Numerical Results
  12. Conclusion

Key Result

Theorem 1

Let $\{\mu^{(k)}\}_{k=1}^\infty$ be the sequence of probability measures generated by eq:wasser_prox, and suppose for all $k$ the step sizes satisfy Then $\mu^{(k)}$ converges to a measure $\mu^*$ in the sense that such that

Figures (4)

  • Figure 1: Error between the capacity $C$ and the computed values $\hat{C}$ vs. input power $P$ for different number of particles.
  • Figure 2: Resulting particles (blue) vs. theoretical optimal input distributions (red).
  • Figure 3: Results for fading channel with and without csir.
  • Figure 4: Evolution of particles over update steps and final histogram for fading channel without csir, $P=1$.

Theorems & Definitions (3)

  • Theorem 1
  • proof
  • Theorem 2