An Optimal Transport Approach to the Computation of the LM Rate

TL;DR

This paper introduces an optimal transport-based method utilizing the Sinkhorn algorithm to efficiently compute the LM rate, a key metric in mismatch capacity, overcoming previous computational challenges.

Contribution

It reformulates the LM rate computation as an optimal transport problem, enabling fast and accurate solutions with minimal iterations.

Findings

OT approach is computationally efficient.

Few Sinkhorn iterations suffice for convergence.

Method effectively handles large input alphabets.

Abstract

Mismatch capacity characterizes the highest information rate for a channel under a prescribed decoding metric, and is thus a highly relevant fundamental performance metric when dealing with many practically important communication scenarios. Compared with the frequently used generalized mutual information (GMI), the LM rate has been known as a tighter lower bound of the mismatch capacity. The computation of the LM rate, however, has been a difficult task, due to the fact that the LM rate involves a maximization over a function of the channel input, which becomes challenging as the input alphabet size grows, and direct numerical methods (e.g., interior point methods) suffer from intensive memory and computational resource requirements. Noting that the computation of the LM rate can also be formulated as an entropy-based optimization problem with constraints, in this work, we transform the task into an optimal transport (OT) problem with an extra constraint. This allows us to efficiently and accurately accomplish our task by using the well-known Sinkhorn algorithm. Indeed, only a few iterations are required for convergence, due to the fact that the formulated problem does not contain additional regularization terms. Moreover, we convert the extra constraint into a root-finding procedure for a one-dimensional monotonic function. Numerical experiments demonstrate the feasibility and efficiency of our OT approach to the computation of the LM rate.