Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Posterior Matching over Binary-Input Memoryless Symmetric Channels: Non-Asymptotic Bounds and Low-Complexity Encoding

Recep Can Yavas

Abstract

We study variable-length feedback (VLF) codes over binary-input memoryless symmetric (BMS) channels using posterior matching with small-enough-difference (SED) partitioning. Prior analyses of SED-based schemes rely on bounded log-likelihood ratio (LLR) increments, restricting their scope to discrete-output channels such as the binary symmetric channel (BSC). We remove this restriction and provide an analysis of posterior matching that covers a broad class of BMS channels, including continuous-output channels such as the binary-input AWGN channel. We derive a novel non-asymptotic achievability bound on the expected decoding time that decomposes into communication, confirmation, and recovery terms with explicit dependence on the channel capacity~$C$, the KL divergence~$C_1$, and the Bhattacharyya parameter of the channel. The proof develops new stopping-time and overshoot bounds for submartingales and random walks with unbounded increments, drawing on tools from renewal theory. On the algorithmic side, we propose a low-complexity encoder that enforces the exact SED partition at every step by grouping messages according to their log-likelihood ratios that are assumed to land on a lattice, and applying a batched correction step that restores the partition balance. The resulting encoder complexity is polynomial in the number of transmitted bits. For continuous-output channels, the lattice structure is enforced through output quantization satisfying an exact induced-lattice constraint; the associated capacity loss is $O(\log B / B^2)$ for a $B$-level quantizer. These results yield a VLF coding scheme for BMS channels that simultaneously achieves strong non-asymptotic performance and practical encoder complexity.

Posterior Matching over Binary-Input Memoryless Symmetric Channels: Non-Asymptotic Bounds and Low-Complexity Encoding

Abstract

We study variable-length feedback (VLF) codes over binary-input memoryless symmetric (BMS) channels using posterior matching with small-enough-difference (SED) partitioning. Prior analyses of SED-based schemes rely on bounded log-likelihood ratio (LLR) increments, restricting their scope to discrete-output channels such as the binary symmetric channel (BSC). We remove this restriction and provide an analysis of posterior matching that covers a broad class of BMS channels, including continuous-output channels such as the binary-input AWGN channel. We derive a novel non-asymptotic achievability bound on the expected decoding time that decomposes into communication, confirmation, and recovery terms with explicit dependence on the channel capacity~, the KL divergence~, and the Bhattacharyya parameter of the channel. The proof develops new stopping-time and overshoot bounds for submartingales and random walks with unbounded increments, drawing on tools from renewal theory. On the algorithmic side, we propose a low-complexity encoder that enforces the exact SED partition at every step by grouping messages according to their log-likelihood ratios that are assumed to land on a lattice, and applying a batched correction step that restores the partition balance. The resulting encoder complexity is polynomial in the number of transmitted bits. For continuous-output channels, the lattice structure is enforced through output quantization satisfying an exact induced-lattice constraint; the associated capacity loss is for a -level quantizer. These results yield a VLF coding scheme for BMS channels that simultaneously achieves strong non-asymptotic performance and practical encoder complexity.

Paper Structure

This paper contains 67 sections, 22 theorems, 217 equations, 4 figures, 1 table, 5 algorithms.

Table of Contents

  1. Introduction
  2. Main Contributions
  3. Paper Organization
  4. Notation and Definitions
  5. Problem Formulation and Preliminaries
  6. SED Scheme and Its Variants
  7. Main Result
  8. Comparison with the Literature
  9. Analysis of the SED Scheme over Arbitrary BMS Channels
  10. Supporting Results on Stopping Times and Ruin Probabilities
  11. Analysis of Error Probability and Expected Stopping Time
  12. Error probability
  13. Score process and stopping rule
  14. Reduction to a uniform per-message bound
  15. Three-phase decomposition
  16. ...and 52 more sections

Key Result

Theorem 1

Fix a positive integer $M$ and error probability $\epsilon \in (0, \frac{1}{2})$. Fix a BMS channel $P_{Y|X}$ with capacity $C > 0$, KL divergence $C_1 = D(P_+ \| P_-) < \infty$An important example where $C_1 = \infty$ is the BEC. In that case, a simple coding scheme that retransmits each informatio where $\eta_0^-(P_{\Lambda})$ and $\eta(P_{\Lambda})$ are given in eq:eta0negdef--eq:etadef evaluat

Figures (4)

  • Figure 1: Rate vs. expected decoding time, BSC$(0.11)$, $\epsilon = 10^{-3}$.
  • Figure 2: Rate vs. expected decoding time, BI-AWGN ($\sigma^2 = 1$, $B = 31$), $\epsilon = 10^{-3}$.
  • Figure 3: Rate vs. number of quantization levels $B$, BI-AWGN ($\sigma^2 = 1$, $\log_2 M = 40$, $\epsilon = 10^{-3}$).
  • Figure 4: Capacity loss $C_{\mathrm{AWGN}} - C_{B,\delta^*}$ (in nats) of the exact-lattice BI-AWGN quantizer ($\sigma^2 = 1$) as a function of the number of output levels $B$.

Theorems & Definitions (31)

  • Definition 1: BMS channel
  • Remark 1: BMS symmetry of the tilted LLR
  • Definition 2: VLF code polyanskiy2011feedback
  • Theorem 1: Achievability bound for BMS channels
  • Corollary 1: Asymptotic expansion (finite-$\epsilon$ regime)
  • Remark 2
  • Remark 3: Parameters for the BSC and BI-AWGN
  • Theorem 2: Submartingale stopping-time bound
  • Theorem 3: First-passage time of an i.i.d. random walk lorden1970, mogul1974, tchamkertenBurnashev
  • Theorem 4: Spitzer's identity and Lundberg's bound
  • ...and 21 more