# Divergence Scaling of Fixed-Length, Binary-Output, One-to-One Distribution Matching

## TL;DR

This paper introduces an optimal fixed-length binary distribution matcher that minimizes divergence from a target distribution, providing bounds and showing near-optimal performance of existing methods.

## Contribution

It proposes a new distribution matcher optimized for binary outputs and derives bounds demonstrating its near-optimality compared to existing methods.

## Key findings

- The proposed matcher minimizes unnormalized divergence.
- Bounds on divergence grow logarithmically with block length.
- Existing constant composition matchers perform close to the theoretical minimum.

## Abstract

Distribution matching is the process of invertibly mapping a uniformly distributed input sequence onto sequences that approximate the output of a desired discrete memoryless source. The special case of a binary output alphabet and one-to-one mapping is studied. A fixed-length distribution matcher is proposed that is optimal in the sense of minimizing the unnormalized informational divergence between its output distribution and a binary memoryless target distribution. Upper and lower bounds on the unnormalized divergence are computed that increase logarithmically in the output block length $n$. It follows that a recently proposed constant composition distribution matcher performs within a constant gap of the minimal achievable informational divergence.