New Second-Order Achievability Bounds for Coding with Side Information via Type Deviation Convergence

TL;DR

This paper introduces type deviation convergence as a unified, tractable framework to derive second-order achievability bounds in network information theory. By coupling random sequences across blocklengths and using general constant-composition (GCC) channels alongside the Poisson matching lemma, it yields tighter second-order dispersions for Wyner-Ziv, Heegard-Berger, and Gelfand-Pinsker with cost, while reproducing classical results in lossy source and channel coding. The approach delivers improved dispersion terms, offers a consistent workflow across problems, and extends to indirect/ no-side-information scenarios and broadcast channels. The framework promises simpler, sharper finite-blocklength analyses and better insights into how side information and cost constraints affect achievable rates at finite n.

Abstract

We propose a framework for second-order achievability, called type deviation convergence, that is generally applicable to settings in network information theory, and is especially suitable for lossy source coding and channel coding with cost. We give a second-order achievability bound for lossy source coding with side information at the decoder (Wyner-Ziv problem) that improves upon all known bounds (e.g., Watanabe-Kuzuoka-Tan, Yassaee-Aref-Gohari and Li-Anantharam). We also give second-order achievability bounds for lossy compression where side information may be absent (Heegard-Berger problem) and channels with noncausal state information at the encoder and cost constraint (Gelfand-Pinsker problem with cost) that improve upon previous bounds.

Figures (3)

• Figure 1: Left: Illustration for lossy source coding with i.i.d. $X^{(n)},Y^{(n)}$. The three red dots are drawn from a Gaussian distribution with covariance matrix given by the first term in (\ref{['eq:sc_gauss']}) (red ellipse is a contour of the Gaussian distribution), and the blue dots are the red dots plus a Gaussian vector with covariance matrix given by the second term in (\ref{['eq:sc_gauss']}). Right: The optimal scheme where we control the deviation of the type of $Y^{(n)}$ according to the type of $X^{(n)}$, moving the red dots to the blue dots along the blue line.
• Figure 2: Left: Illustration for Wyner-Ziv coding with i.i.d. $X^{(n)},U^{(n)}$. The red dots, blue dots and green dots are samples of $\mathbb{E}[[\iota,d]^{\top}|X]$, $\mathbb{E}[[\iota,d]^{\top}|X,U]$ and $[\iota,d]^{\top}$, respectively (the 3 stages in Section \ref{['subsec:wz_statement']}). Right: The optimal scheme where we control the deviation of the type of $U^{(n)}$ according to the type of $X^{(n)}$, moving the red dots to the blue dots along the blue curve.
• Figure 3: Our upper bound $\mathsf{V}_{\mathrm{GCC}}$ on $\mathsf{V}^{*}(\mathsf{D})$ for binary-Hamming Wyner-Ziv, and previous upper bounds $\mathsf{V}_{\mathrm{VYAG}}$, $\mathsf{V}_{\mathrm{WKT}}$, $\mathsf{V}_{\mathrm{LA}}$ for $p\in\{1/10,1/5,2/5\}$ and $\mathsf{D}\in[0,p]$.

Theorems & Definitions (24)

• Definition 1: Gaussian-multinomial distribution
• Definition 2: Conditional Gaussian-multinomial distribution
• Proposition 3
• Definition 4: Type deviation convergence
• Proposition 5
• Proposition 6
• Proposition 7: Type deviation convergence of memoryless channels
• Proposition 8
• Remark 9
• Definition 10: General constant-composition (GCC) channel