Sparse Regression LDPC Codes

TL;DR

An efficient low-dimensional approximate state evolution recursion that can be used for efficient hyperparameter tuning, thus paving the way for future work on optimal code design is proposed.

Abstract

This article introduces a novel concatenated coding scheme called sparse regression LDPC (SR-LDPC) codes. An SR-LDPC code consists of an outer non-binary LDPC code and an inner sparse regression code (SPARC) whose respective field size and section sizes are equal. For such codes, an efficient decoding algorithm is proposed based on approximate message passing (AMP) that dynamically shares soft information between inner and outer decoders. This dynamic exchange of information is facilitated by a denoiser that runs belief propagation (BP) on the factor graph of the outer LDPC code within each AMP iteration. It is shown that this denoiser falls within the class of non-separable pseudo-Lipschitz denoising functions and thus that state evolution holds for the proposed AMP-BP algorithm. Leveraging the rich structure of SR-LDPC codes, this article proposes an efficient low-dimensional approximate state evolution recursion that can be used for efficient hyperparameter tuning, thus paving the way for future work on optimal code design. Finally, numerical simulations demonstrate that SR-LDPC codes outperform contemporary codes over the AWGN channel for parameters of practical interest. SR-LDPC codes are shown to be viable means to obtain shaping gains over the AWGN channel.

Figures (10)

• Figure 1: This notional diagram depicts the encoding process for an SR-LDPC code. Information message $\mathbf{w}$ is first outer encoded using an LDPC code over $\mathbb{F}_q$. Every LDPC-encoded field element is subsequently converted into a one-sparse basis vector. The collection of one-sparse vectors are then stacked into a SPARC-like sequence. The resulting $L$-sparse vector is pre-multiplied by matrix $\mathbf{A}$. The outcome of this process is the signal $\mathbf{x}$.
• Figure 2: This diagram depicts the operation of the dynamic AMP-BP decoder. The input comes in the form of observation $\mathbf{y}$ at the top left. During every AMP iteration, the algorithm computes the residual $\mathbf{z}$, which incorporates the effect of the Onsager term. This vector is then turned into an effective observation, which acts as the input to the BP denoiser. After message passing, a state estimate vector is produced for every section. This, in turn, yields the updated global estimate via concatenation. The computation of the Onsager term, which is intrinsic to AMP, is highlighted on the left. The iterative process repeats itself until convergence is achieved, at which point the state estimate $\hat{\mathbf{s}}$ is taken as the output of the algorithm.
• Figure 3: This illustration shows the augmented factor graph for the denoising function with the variable nodes, the parity check constraints, and the extra factors associated with local observations. The effective observation vector is sectionized in a way that matches variable nodes.
• Figure 4: This figure compares the true $\tau_t^2$ values with the estimated $\hat{\tau}_t^2$ values predicted by the approximate state evolution algorithm defined in Definition \ref{['definition:approximate_state_evolution']}. The approximate state evolution algorithm is very accurate for low and high SNRs but provides reduced insight when AMP is at the edge of its convergence region.
• Figure 5: This figure demonstrates the utility of the approximate state evolution algorithm for hyperparameter optimization by comparing $\tau_T^2 - \sigma^2$ for SR-LDPC codes with a constant overall rate but a varying inner LDPC code rate. The approximate state evolution tool may be used as a coarse optimization tool, possibly followed by fine tuning through a local rate search.

Theorems & Definitions (36)

• Remark 1
• Definition 3: Vector $+g$ Operator bennatan2006design
• Definition 4: Vector $\times g$ Operator bennatan2006design
• Remark 5
• Definition 6: BP Denoiser
• Theorem 8: BP Denoiser is Lipschitz Continuous
• Proposition 9
• Definition 10: Geometric Uniformity forney1991geometrically
• Proposition 11
• Proposition 12