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The Error Probability of Spatially Coupled Sparse Regression Codes over Memoryless Channels

Yuhao Liu, Yizhou Xu, Tianqi Hou

TL;DR

This work provides a rigorous non-asymptotic analysis of SC-SPARCs over general memoryless channels using the GAMP decoder. By characterizing the SE-driven decoding progression and establishing a non-asymptotic concentration bound, it proves that the section error probability decays exponentially with code length $n$ whenever the rate is below the channel capacity ($R<\mathcal{C}$). The analysis hinges on a traveling wave decoding mechanism under spatial coupling and a carefully designed base matrix with row normalization, yielding a finite-length performance guarantee for SC-SPARCs beyond the AWGN setting. Consequently, the paper advances the theoretical understanding of capacity-achieving SPARCs on generic memoryless channels and motivates practical explorations of SC-SPARCs with robust non-asymptotic guarantees and broader channel models.

Abstract

Sparse Regression Codes (SPARCs) are capacity-achieving codes introduced for communication over the Additive White Gaussian Noise (AWGN) channels and were later extended to general memoryless channels. In particular it was shown via threshold saturation that Spatially Coupled Sparse Regression Codes (SC-SPARCs) are capacity-achieving over general memoryless channels when using an Approximate Message Passing decoder (AMP). This paper, for the first time rigorously, analyzes the non-asymptotic performance of the Generalized Approximate Message Passing (GAMP) decoder of SC-SPARCs over memoryless channels, and proves exponential decaying error probability with respect to the code length.

The Error Probability of Spatially Coupled Sparse Regression Codes over Memoryless Channels

TL;DR

This work provides a rigorous non-asymptotic analysis of SC-SPARCs over general memoryless channels using the GAMP decoder. By characterizing the SE-driven decoding progression and establishing a non-asymptotic concentration bound, it proves that the section error probability decays exponentially with code length whenever the rate is below the channel capacity (). The analysis hinges on a traveling wave decoding mechanism under spatial coupling and a carefully designed base matrix with row normalization, yielding a finite-length performance guarantee for SC-SPARCs beyond the AWGN setting. Consequently, the paper advances the theoretical understanding of capacity-achieving SPARCs on generic memoryless channels and motivates practical explorations of SC-SPARCs with robust non-asymptotic guarantees and broader channel models.

Abstract

Sparse Regression Codes (SPARCs) are capacity-achieving codes introduced for communication over the Additive White Gaussian Noise (AWGN) channels and were later extended to general memoryless channels. In particular it was shown via threshold saturation that Spatially Coupled Sparse Regression Codes (SC-SPARCs) are capacity-achieving over general memoryless channels when using an Approximate Message Passing decoder (AMP). This paper, for the first time rigorously, analyzes the non-asymptotic performance of the Generalized Approximate Message Passing (GAMP) decoder of SC-SPARCs over memoryless channels, and proves exponential decaying error probability with respect to the code length.
Paper Structure (12 sections, 9 theorems, 64 equations, 1 figure, 1 algorithm)

This paper contains 12 sections, 9 theorems, 64 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Settings
  3. Code Structure
  4. GAMP decoder for SC-SPARCs
  5. Decoding Progression According to SE
  6. Non-asymptotic performance of the GAMP decoder for SC-SPARC
  7. Future Work
  8. Proof of propositions
  9. Proof of Proposition \ref{['prop:prop1']}
  10. Proof of Proposition \ref{['prop:prop2']}
  11. Concentration on the SE
  12. Performance of SC-GAMP algorithm for GLM

Key Result

Proposition 1

For a continuously differentiable conditional probability density function $P_{\text{out}} (y | x)$ with respect to $x$, the expression of $f_{\text{out}}(\sigma)$ in (eq:fout1) is where $P^{\prime}_{\text{out}} (y|x) = \frac{\partial}{\partial x} P_{\text{out}} (y|x)$. Then, $f_{\text{out}} (\sigma)$ is non-negative for all $\sigma \in [0,1]$.

Figures (1)

  • Figure 1: A spatially coupled coding matrix for SC-SPARCs with parameters $\omega = 2, \Gamma = 12$. The matrix $\mathbf{A}$ is partitioned to $\Gamma \times \Gamma$ blocks, indexed by $(r,c)$ and labeled as $\mathbf{A}_{rc}$. Each of these block comprises $N / \Gamma$ columns and $n/\Gamma = \alpha N/\Gamma$ rows where $\alpha = (\log M) / MR$. The i.i.d elements in $\mathbf{A}_{rc}$ are distributed as $\mathcal{N}(0, W_{rc} / L)$. In each row, variances in blue blocks collectively share the power $(1-\rho)\Gamma$ equally, while the variances in yellow blocks share the power $\rho \Gamma$. This meticulous design satisfies the row normalization condition $\frac{1}{\Gamma} \sum_{c} W_{rc} = 1, \forall r$. The design matrix degrades to a special case of SC ensembles in barbier2019universal when $\rho=0$.

Theorems & Definitions (13)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • proof
  • Lemma 2
  • Lemma 3
  • ...and 3 more