DeepPolar: Inventing Nonlinear Large-Kernel Polar Codes via Deep Learning

TL;DR

DeepPolar codes extend the conventional Polar coding framework by utilizing a larger kernel size and parameterizing these kernels and matched decoders through neural networks, resulting in enhanced reliability when compared to both existing neural codes and conventional Polar codes.

Abstract

Progress in designing channel codes has been driven by human ingenuity and, fittingly, has been sporadic. Polar codes, developed on the foundation of Arikan's polarization kernel, represent the latest breakthrough in coding theory and have emerged as the state-of-the-art error-correction code for short-to-medium block length regimes. In an effort to automate the invention of good channel codes, especially in this regime, we explore a novel, non-linear generalization of Polar codes, which we call DeepPolar codes. DeepPolar codes extend the conventional Polar coding framework by utilizing a larger kernel size and parameterizing these kernels and matched decoders through neural networks. Our results demonstrate that these data-driven codes effectively leverage the benefits of a larger kernel size, resulting in enhanced reliability when compared to both existing neural codes and conventional Polar codes.

Figures (24)

• Figure 1: DeepPolar ($n$=256, $k$=37) with appropriate kernel sizes (e.g., $\ell=16,32$) outperforms classical Reed-Muller, Polar, and state-of-the-art neural KO codes Makkuva2021 on AWGN channels with -2dB SNR
• Figure 2: Channel coding via deep learning
• Figure 3: (a) Polar$(4,3)$ encoding structure using the standard $2 \times 2$ kernel. Encoding is performed recursively on the Plotkin tree. (b) Polar$(16,8)$ encoding using $4\times 4$ kernels. (c) DeepPolar($4,3,\ell=2)$ replaces the xor operation in Plotkin - $2 \times 2$ by neural networks. (d) DeepPolar($16,8,\ell=4)$ : Scaling the DeepPolar encoding to higher-order kernels enables us to achieve good reliability. We are the first to explore this design space
• Figure 4: The curriculum to train DeepPolar$(16,8,\ell=4)$ proceeds in two phases: (a) In Phase 1, the kernels $(4,i)$ are trained progressively for $i = 1,\cdots, \ell$. (b) In phase 2, we initialize each kernel in the encoder by the respective kernels (and similarly for the decoders). For instance, the kernel $g_{11}$ has one information input bit; we initialize it with a pretrained $g_\phi(4,1)$. Similarly, $g_{20}$ has three information input bit-groups; we initialize it with a pretrained $g_\phi(4,3)$.
• Figure 5: (a) DeepPolar improves over state-of-the-art KO codes Makkuva2021, and RM, Polar codes at $n$=256,$k$=37. (b,c) DeepPolar is suitable for a variety of rates and retains gains over Polar, while KO is not suitable for these rates.