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Capacity-achieving sparse superposition codes with spatially coupled VAMP decoder

Yuhao Liu, Teng Fu, Jie Fan, Panpan Niu, Chaowen Deng, Zhongyi Huang

TL;DR

This work proposes a spatially coupled VAMP (SC-VAMP) decoder for SS with spatially coupled design matrices and shows that the SC-VAMP decoder outperforms the VAMP decoder with exponential decay power allocation, achieving a lower section error rate.

Abstract

Sparse superposition (SS) codes provide an efficient communication scheme over the Gaussian channel, utilizing the vector approximate message passing (VAMP) decoder for rotational invariant design matrices. Previous work has established that the VAMP decoder for SS achieves Shannon capacity when the design matrix satisfies a specific spectral criterion and exponential decay power allocation is used. In this work, we propose a spatially coupled VAMP (SC-VAMP) decoder for SS with spatially coupled design matrices. Based on state evolution (SE) analysis, we demonstrate that the SC-VAMP decoder is capacity-achieving when the design matrices satisfy the spectra criterion. Empirically, we show that the SC-VAMP decoder outperforms the VAMP decoder with exponential decay power allocation, achieving a lower section error rate. All codes are available on https://github.com/yztfu/SC-VAMP-for-Superposition-Code.git.

Capacity-achieving sparse superposition codes with spatially coupled VAMP decoder

TL;DR

This work proposes a spatially coupled VAMP (SC-VAMP) decoder for SS with spatially coupled design matrices and shows that the SC-VAMP decoder outperforms the VAMP decoder with exponential decay power allocation, achieving a lower section error rate.

Abstract

Sparse superposition (SS) codes provide an efficient communication scheme over the Gaussian channel, utilizing the vector approximate message passing (VAMP) decoder for rotational invariant design matrices. Previous work has established that the VAMP decoder for SS achieves Shannon capacity when the design matrix satisfies a specific spectral criterion and exponential decay power allocation is used. In this work, we propose a spatially coupled VAMP (SC-VAMP) decoder for SS with spatially coupled design matrices. Based on state evolution (SE) analysis, we demonstrate that the SC-VAMP decoder is capacity-achieving when the design matrices satisfy the spectra criterion. Empirically, we show that the SC-VAMP decoder outperforms the VAMP decoder with exponential decay power allocation, achieving a lower section error rate. All codes are available on https://github.com/yztfu/SC-VAMP-for-Superposition-Code.git.

Paper Structure

This paper contains 6 sections, 1 theorem, 19 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Spatially coupled SS construction
  3. SC-VAMP-based decoder for SC-SS codes
  4. State evolution analysis
  5. Numerical results
  6. Further Work

Key Result

Proposition 1

Consider the limit SE with the same $F_{\textsf{r}}(x)=F(x)$ and initialization $\sigma^0_{\textsf{r}} =1$ for $\textsf{r} \in [\textsf{R}]$. We define $R_{\textnormal{alg}} := F(1) / 2$ and $R_{\textnormal{IT}} := \int_{0}^{1} F(x) \mathrm{d} x / 2$. Then we have

Figures (4)

  • Figure 1: A SC-SS with spatial coupling parameters $\Gamma = 5$ and $W = 2$, thus $\textsf{R} = 6$, $\textsf{C} = 5$ and $\vartheta = 6/5$. Each $\bm{x}_{\textsf{c}}$ consists of $L/\Gamma$ for $\textsf{c} \in [\textsf{C}]$, and each $\vec{\bm{x}}\mkern3mu_{\textsf{r}}$ consists of $|W_{\textsf{r}}| L / \Gamma$ sections for $\textsf{r} \in [\textsf{R}]$. Each design matrix $\bm{A}_{\textsf{r}}$ contains $M/(\Gamma+W-1)$ rows and $N|W_{\textsf{r}}|/ \Gamma$ columns for $\textsf{r} \in [\textsf{R}]$.
  • Figure 2: Factor graph representation of the joint probability distribution of $\{\bm{x}_{\textsf{c}}\}_{\textsf{c}}^{\textsf{C}}$ and $\{ \vec{\bm{x}}\mkern3mu_{\textsf{r}}\}_{\textsf{r}}^{\textsf{R}}$, with spatial coupling parameters $\Gamma = 5, W=2$ and $\vartheta = 6/5$. The function factors $\{ p_{\textsf{c}}(\cdot)\}_{\textsf{c}}^{\textsf{C}}$ represent $P_0^{\otimes(L/\textsf{C})}$, and the factors $\{ \delta_{\textsf{r}}(\cdot)\}_{\textsf{r}}^{\textsf{R}}$ represent the Dirac delta distributions.
  • Figure 3: Overall section error rate $\text{SER}^k = (\sum_{\textsf{c}=1}^\textsf{C} \text{SER}^{k}_{\textsf{c}} )/ \textsf{C}$ ran on a random single instance with $L = 2^{14}$, $B=16$ and $\text{snr} = 15$, as a function of the iterations. For SC-VAMP decoder, we set $\Gamma = 16$, $W = 2$ and $\vartheta = 17/16$. The overall rate $R_{\text{all}} = 1.60$ is higher than the algorithm threshold, lower than the information-theoretic threshold proposed in hou2022sparse.
  • Figure 4: $\text{SER}_{\textsf{c}}^k$ vs. block index $\textsf{c} \in [\textsf{C}]$ for several iteration numbers in a SC-VAMP decoding process, using the same setup as in Fig.\ref{['fig:decoder-SER']}. The solid line corresponds to DCT matrices, while the dotted line corresponds to Gaussian matrices.

Theorems & Definitions (2)

  • Proposition 1
  • proof