Convergence of Weighted Min-Sum Decoding Via Dynamic Programming on Trees

• Published in: IEEE transactions on information theory Vol. 60; no. 2; pp. 943 - 963
• Main Authors: Yung-Yih Jian, Pfister, Henry D.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.02.2014; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithm design and analysis; Algorithms; Applied sciences; Belief propagation; Codes; Coding, codes; Convergence; Decoding; Error correcting codes; Exact sciences and technology; Information theory; Information, signal and communications theory; LDPC codes; Linear programming; Low density parity check codes; LP decoding; Mathematical models; max-product algorithm; Maximum likelihood decoding; Maximum likelihood method; min-sum algorithm; Noise; Operations research; Parity check codes; Signal and communications theory; Telecommunications and information theory; Thresholds; Vectors; Performance evaluation; max-product algorithm; min-sum algorithm; Linear programming; LP decoding; Algorithm; Computational complexity; Counterexample; LDPC codes; Maximum likelihood decoding; Credal approach; Error correcting code; Parity check codes; Dynamic programming; Fixed point; Belief propagation
• ISSN: 0018-9448; 1557-9654; 1557-9654
• DOI: 10.1109/TIT.2013.2290303

Applying the max-product (and sum-product) algorithms to loopy graphs is now quite popular for best assignment problems. This is largely due to their low computational complexity and impressive performance in practice. Still, there is no general understanding of the conditions required for convergence or optimality of converged solutions or both. This paper presents an analysis of both attenuated max-product decoding and weighted min-sum decoding for low-density parity-check (LDPC) codes, which guarantees convergence to a fixed point when a weight factor, β, is sufficiently small. It also shows that, if the fixed point satisfies some consistency conditions, then it must be both a linear-programming (LP) and maximum-likelihood (ML) decoding solution. For (d v , d c )-regular LDPC codes, the weight factor must satisfy β(d v -1) <; 1 to guarantee convergence to a fixed point, whereas the results proposed by Frey and Koetter require instead that β(d v -1)(d c -1) ≤ 1. In addition, the range of the weight factor for a provable ML decoding solution is extended to 0 <; β(d v -1) 1. In addition, counterexamples that show a fixed point might not be the ML decoding solution if β(d v -1) > 1 are given. Finally, connections are explored with recent work on the threshold of LP decoding.