Soft Covering with High Probability

TL;DR

This paper strengthens Wyner's soft-covering lemma by demonstrating that a random codebook achieves soft covering with high probability, with failure probability decreasing super-exponentially, thus broadening its applicability in information theory.

Contribution

It moves from expected value analysis to high-probability guarantees, providing bounds on decay rates and second-order rates for soft covering.

Findings

Super-exponentially small failure probability

High-probability soft-covering achieved by random codebooks

Bounds on decay rate and second-order codebook rate

Abstract

Wyner's soft-covering lemma is the central analysis step for achievability proofs of information theoretic security, resolvability, and channel synthesis. It can also be used for simple achievability proofs in lossy source coding. This work sharpens the claim of soft-covering by moving away from an expected value analysis. Instead, a random codebook is shown to achieve the soft-covering phenomenon with high probability. The probability of failure is super-exponentially small in the block-length, enabling many applications through the union bound. This work gives bounds for both the exponential decay rate of total variation and the second-order codebook rate for soft covering.