Gradient Flow Decoding

TL;DR

This paper introduces Gradient Flow decoding for LDPC codes, a continuous-time approach based on gradient flow that is suitable for analog implementation and performs comparably to existing algorithms in various channel conditions.

Contribution

It presents a novel gradient flow decoding method for LDPC codes, including a generalization using the negative log-likelihood function and its tensor-computability for AI accelerators.

Findings

Performance comparable to multi-bit gradient descent bit flipping.

Generalized GF decoding rivals MMSE + BP in LDPC-MIMO channels.

Method is tensor-computable and suitable for AI hardware.

Abstract

This paper presents the Gradient Flow (GF) decoding for LDPC codes. GF decoding, a continuous-time methodology based on gradient flow, employs a potential energy function associated with bipolar codewords of LDPC codes. The decoding process of the GF decoding is concisely defined by an ordinary differential equation and thus it is well suited to an analog circuit implementation. We experimentally demonstrate that the decoding performance of the GF decoding for AWGN channels is comparable to that of the multi-bit mode gradient descent bit flipping algorithm. We further introduce the negative log-likelihood function of the channel for generalizing the GF decoding. The proposed method is shown to be tensor-computable, which means that the gradient of the objective function can be evaluated with the combination of basic tensor computations. This characteristic is well-suited to emerging AI accelerators, potentially applicable in wireless signal processing. The paper assesses the decoding performance of the generalized GF decoding in LDPC-coded MIMO channels. Our numerical experiments reveal that the decoding performance rivals that of established techniques like MMSE + BP. Furthermore, an exploration of score-based channel learning for capturing statistical properties is also provided.