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Scaling Deep Learning-based Decoding of Polar Codes via Partitioning
Sebastian Cammerer, Tobias Gruber, Jakob Hoydis, Stephan ten Brink
TL;DR
Deep-learning channel decoding is constrained by exponential training complexity as codebooks grow. This paper partitions polar decoding into small neural sub-blocks connected by conventional belief propagation, achieving competitive BER with non-iterative decoding, while partitioning still limits performance and code length.
Problem
Exponential training complexity limits neural-network decoding to very short polar-code block lengths, despite its low-latency and near-MAP advantages.
Method
The encoding graph is partitioned into small sub-blocks whose decoders are trained independently as neural networks and coupled through remaining conventional BP stages.
Results
The resulting decoder enables non-iterative decoding with competitive BER performance relative to conventional SC and BP decoding.
Takeaways & Limitations
Replacing decoder sub-components with neural networks improves scalability toward practical lengths while retaining a structure amenable to parallel implementation.
Takeaways & Limitations
Performance degradation is mainly caused by small partitions, because training currently limits sub-block size and therefore restricts the approach to short codes.
Abstract
from arXiv · showhide
The training complexity of deep learning-based channel decoders scales exponentially with the codebook size and therefore with the number of information bits. Thus, neural network decoding (NND) is currently only feasible for very short block lengths. In this work, we show that the conventional iterative decoding algorithm for polar codes can be enhanced when sub-blocks of the decoder are replaced by neural network (NN) based components. Thus, we partition the encoding graph into smaller sub-blocks and train them individually, closely approaching maximum a posteriori (MAP) performance per sub-block. These blocks are then connected via the remaining conventional belief propagation decoding stage(s). The resulting decoding algorithm is non-iterative and inherently enables a high-level of parallelization, while showing a competitive bit error rate (BER) performance. We examine the degradation through partitioning and compare the resulting decoder to state-of-the-art polar decoders such as successive cancellation list and belief propagation decoding.
I. INTRODUCTION
Deep-learning decoders offer non-iterative, low-latency operation and near-MAP performance, but exponential training complexity limits practical block lengths. The paper addresses this by replacing only polar-decoder sub-components with independently trained neural networks.
- Deep-learning decoding provides non-iterative, low-latency operation and close-to-MAP performance.
- Learning all 2^k codewords creates exponential training complexity as the number of information bits increases.
- Polar codes offer strong short-block BER performance and flexible code rates but have inherently serial decoding that is difficult to parallelize.
- The proposed partitioned approach aims to extend neural decoding toward practical block lengths while retaining static implementations suitable for parallel hardware.
- Unlike approaches that only refine belief-propagation weights while retaining iterative decoding, this work replaces polar-decoder components with neural networks.
II. POLAR CODES
Polar coding maps information and frozen bits through a structured encoding graph that also supports belief-propagation decoding. The decoder exchanges soft LLR messages through iterative left-to-right and right-to-left propagations.
- A polar encoder places k information bits in the most reliable positions of u and treats the remaining N−k positions as known frozen bits.
- The encoding circuit forms a decoding graph with log2(N)+1 stages, each containing N nodes.
- Polar decoding follows either serial successive-cancellation methods or iterative belief-propagation message passing.
- This work uses belief propagation because its graph structure matches neural networks and supports parallel processing.
- Each BP iteration updates soft LLR messages first from the a priori-information stage toward channel information, then in the reverse direction.
- For N=8, Fig. 1 marks independent partitions with blue boxes for M=2 and green boxes for M=4.
A. Partitionable Codes
Polar codes’ regular encoding graph can be partitioned into independently decodable sub-graphs. The proposed approach replaces these sub-decoders with neural networks and couples them through the remaining polar stages.
- A. Partitionable Codes: Polar encoding graphs contain independent sub-graphs because their first log2(Np) stages have no interconnections.
- A. Partitionable Codes: A partitionable code allows each sub-block to be decoded independently, with coupling occurring only through the remaining polar stages.
- A. Partitionable Codes: Frozen positions are divided among sub-blocks, so different sub-blocks may have different code rates.
- A. Partitionable Codes: The proposed method replaces conventional single-parity-check, repetition, or other partition sub-decoders with neural networks.
B. Deep-Learning for Channel Coding
Deep-learning channel decoders use feedforward neural networks to map noisy codewords to information-bit probabilities, but training remains constrained by exponentially growing decoding complexity.
- Neural-network decoding: Feedforward neural networks compose layer functions into an input-output mapping with trainable weights Θ.Training uses known input-output mappings with gradient descent and backpropagation.
- Neural-network decoding: The decoder maps noisy codewords to information bits using ReLU hidden layers and a sigmoid output representing the probability that each bit is “1”.
- Performance and scalability: Small polar codes can reach MAP performance with neural-network decoding.The BER gap between NND and MAP is used to assess this performance.
- Performance and scalability: Exponential complexity as the number of information bits increases limits learning-based decoding scalability.The limitation is illustrated by the BER performance gap between NND and MAP decoding.
III. PARTITIONED POLAR NEURAL NETWORK DECODING
The partitioned decoder replaces sequentially trained sub-block decoders with independently trained neural networks, coupling them through conventional belief propagation to form a one-shot decoding process.
- Partitioned decoding: Each sub-block neural decoder covers k_i information bits, making efficient training feasible only when sub-blocks contain few information bits.Each sub-block has its own frozen-bit positions A_i and block length N_i.
- Partitioned decoding: The reduced effective sub-block size enables offline training toward MAP performance, with outputs represented as soft values or hard decisions.Hard decisions may require re-encoding before subsequent processing.
- Decoder structure: The proposed decoder combines M deep-learning blocks for sub-codewords with a conventional BP part that propagates decoded sub-blocks through remaining coupling stages.
- Decoder structure: After BP initialization and message updates, sub-blocks are decoded top-to-bottom, re-encoded, and treated as known frozen bits before continuing.The process repeats until all sub-blocks are decoded.
- One-shot decoding: Because each neural network is passed once, the pipelined architecture is termed one-shot decoding.The complete system can also be modeled as one large neural network with deterministic BP layers.
A. Further Optimizations
Further optimizations reduce partitioning overhead and adapt independently trained components to the coupled decoder, while BP remains the preferred coupling mechanism.
- Partition-size optimization: The number of information bits per sub-block k_i limits neural-network estimates, motivating merging equally sized sub-blocks to reduce the number of partitions.
- System fine-tuning: Fine-tuning the overall network accounts for conventional decoding stages and can prevent degradation from non-Gaussian neural-network input distributions.The issue can arise when LLR clipping affects the assumed input distribution.
- Coupling strategy: BP is preferred over SC for coupling because its graph structure and edge weights better match neural-network representations and support efficient parallel processing.
- CRC-aided decoding: Splitting a whole-codeword CRC into sub-block CRCs is possible, but sufficiently large neural-network sub-blocks are required to avoid prohibitive rate loss.The approach is therefore not considered at this stage.
IV. COMPARISON WITH SCL/BP
The comparison evaluates partitioned NN decoding against conventional polar decoders and separates performance loss from partitioning versus NN suboptimality. It also examines the resulting latency through synchronization-step counts.
- For N = 128, the proposed decoder uses partitions limited to kmax = 12 information bits to facilitate sub-block training.
- PSCL provides a lower bound on expected BER for a given partition count, enabling partitioning and NN losses to be examined separately.
- Partitioning loss increases as the number of partitions grows, while larger code lengths also make MAP-level NN training more difficult.
- The main normalized error originates from partitioning, with only a small contribution from suboptimal neural networks.
- The partitioned NN decoder reduces BP latency through parallel operations while remaining competitive with SC and BP in BER performance.
V. CONCLUSION
Replacing decoder sub-components with neural networks improves scalability toward practical block lengths and enables non-iterative, parallel decoding. However, small trainable partitions still limit code length and cause performance degradation.
- Replacing sub-components of an existing decoder enables scalability in block length and number of information bits toward practical lengths.
- BER performance is similar to SC and BP decoding, while exploiting parallelism enables one-shot, non-iterative decoding with substantially reduced latency.
- Performance degradation mainly results from small partitions because the sub-block size is currently limited by training.