An Optimal Transport-Based Method for Computing LM Rate and Its Convergence Analysis

TL;DR

This paper introduces an efficient optimal transport-based method using the Sinkhorn algorithm to compute the LM rate, a tighter lower bound on mismatch capacity, with proven convergence and practical effectiveness.

Contribution

It reformulates LM rate computation as an optimal transport problem and develops a novel Sinkhorn-based algorithm with convergence analysis.

Findings

The proposed method efficiently computes the LM rate for large input alphabets.

Numerical experiments confirm the algorithm's practicality and convergence.

The approach outperforms traditional numerical methods in computational efficiency.

Abstract

The mismatch capacity characterizes the highest information rate of the channel under a prescribed decoding metric and serves as a critical performance indicator in numerous practical communication scenarios. Compared to the commonly used Generalized Mutual Information (GMI), the Lower bound on the Mismatch capacity (LM rate) generally provides a tighter lower bound on the mismatch capacity. However, the efficient computation of the LM rate is significantly more challenging than that of the GMI, particularly as the size of the channel input alphabet increases. This growth in complexity renders standard numerical methods (e.g., interior point methods) computationally intensive and, in some cases, impractical. In this work, we reformulate the computation of the LM rate as a special instance of the optimal transport (OT) problem with an additional constraint. Building on this formulation, we develop a novel numerical algorithm based on the Sinkhorn algorithm, which is well known for its efficiency in solving entropy regularized optimization problems. We further provide the convergence analysis of the proposed algorithm, revealing that the algorithm has a sub-linear convergence rate. Numerical experiments demonstrate the feasibility and efficiency of the proposed algorithm for the computation of the LM rate.