On the Strong Converse Exponent and Error Exponent of the Classical Soft Covering

TL;DR

The paper resolves the exact strong converse exponent for classical soft covering and introduces a novel two-parameter information quantity $J_{\alpha,\beta}$ to express the exponent $\Gamma(R)$, establishing equality across uniform, non-uniform, and $H_{-\infty}$-constrained formulations. It demonstrates that random coding is not generally tight in this setting and reveals a rational–irrational discrepancy under the conventional uniform formulation, which is remedied by the $H_{-\infty}$-constrained formulation that preserves the low-rate behavior. The work extends the analysis to noiseless and noisy channels, showing high-rate improvements over random coding and providing both achievability and converse bounds, including an infinite exponent scenario when a rational input distribution exists. These results offer deep insights into soft covering performance, influence code-design strategies, and suggest broader implications for related problems in channel resolvability and channel simulation.

Abstract

This paper establishes the exact strong converse exponent of the soft covering problem in the classical setting. This exponent characterizes the slowest achievable convergence speed of the total variation to one when a code of rate below mutual information is applied to a discrete memoryless channel for synthesizing a product output distribution. The proposed exponent is expressed through a new two-parameter information quantity, differing from the more commonly studied Rényi divergence or Rényi mutual information. In addition, we demonstrate the non-tightness of random coding for rates both below and above mutual information. Discussions on the latter start with noiseless channels, where we develop a deterministic code construction that outperforms random codes in error exponents. We further observe that the conventional formulation, which assumes a uniform distribution over messages, inherently introduces a discrepancy in error exponents depending on whether the components of the target distribution are rational or irrational numbers. To eliminate this discrepancy, we propose a new formulation in which messages are allowed to be distributed non-uniformly, and the rate is given by the logarithm of the smallest nonzero message probability (corresponding to Rényi entropy $H_{-\infty}$ of order $-\infty$). The exact error exponent is characterized in this formulation for noiseless channels. Furthermore, for noisy channels, we provide a high-rate improvement in achievability and derive a converse bound on the error exponent.

Figures (5)

• Figure 1: Examples of the exact strong converse exponent $\mathit{\Gamma}(R)$, the random coding achievability $\overline{\mathit{\Gamma}}_{\mathrm{rc}}(R)$, and the random coding converse $\underline{\mathit{\Gamma}}_{\mathrm{rc}}(R)$.
• Figure 2: Examples of error exponents for noiseless channels under the uniform, non-uniform, and $H_{-\infty}$-constrained formulations.
• Figure 3: Examples of the converse bound $\overline{E}(R)$, the achievability bound $\underline{E}(R)$, and the random-coding error exponent $E_{\mathrm{rc}}(R)$ for noisy channels.
• Figure 4: Three possible ways in which $\mathit{\Gamma}(s,Q_X,R)$ intersects $s$. Here, $\mathcal{X}_l$ denotes the set of noiseless input symbols, and $\mathcal{X}_o$ denotes the set of noisy input symbols.
• Figure 5: Examples of the converse bound $\underline{\mathit{\Gamma}}(R)$ and the achievability bound $\overline{\mathit{\Gamma}}(R)$ for the strong converse exponent.

Theorems & Definitions (74)

• definition 1: Information density
• definition 2: Expectation of the information density
• lemma 1
• definition 3: $\alpha$-Rényi entropy
• remark 1
• definition 4: $\alpha$-Rényi divergence
• definition 5: $\alpha$-Rényi mutual information
• remark 2
• definition 6: Uniform soft covering
• definition 7: Non-uniform soft covering