A Low Density Lattice Decoder via Non-Parametric Belief Propagation

📝 Abstract

The recent work of Sommer, Feder and Shalvi presented a new family of codes called low density lattice codes (LDLC) that can be decoded efficiently and approach the capacity of the AWGN channel. A linear time iterative decoding scheme which is based on a message-passing formulation on a factor graph is given. In the current work we report our theoretical findings regarding the relation between the LDLC decoder and belief propagation. We show that the LDLC decoder is an instance of non-parametric belief propagation and further connect it to the Gaussian belief propagation algorithm. Our new results enable borrowing knowledge from the non-parametric and Gaussian belief propagation domains into the LDLC domain. Specifically, we give more general convergence conditions for convergence of the LDLC decoder (under the same assumptions of the original LDLC convergence analysis). We discuss how to extend the LDLC decoder from Latin square to full rank, non-square matrices. We propose an efficient construction of sparse generator matrix and its matching decoder. We report preliminary experimental results which show our decoder has comparable symbol to error rate compared to the original LDLC decoder.%

💡 Analysis

The recent work of Sommer, Feder and Shalvi presented a new family of codes called low density lattice codes (LDLC) that can be decoded efficiently and approach the capacity of the AWGN channel. A linear time iterative decoding scheme which is based on a message-passing formulation on a factor graph is given. In the current work we report our theoretical findings regarding the relation between the LDLC decoder and belief propagation. We show that the LDLC decoder is an instance of non-parametric belief propagation and further connect it to the Gaussian belief propagation algorithm. Our new results enable borrowing knowledge from the non-parametric and Gaussian belief propagation domains into the LDLC domain. Specifically, we give more general convergence conditions for convergence of the LDLC decoder (under the same assumptions of the original LDLC convergence analysis). We discuss how to extend the LDLC decoder from Latin square to full rank, non-square matrices. We propose an efficient construction of sparse generator matrix and its matching decoder. We report preliminary experimental results which show our decoder has comparable symbol to error rate compared to the original LDLC decoder.%

📄 Content

arXiv:0901.3197v4 [cs.IT] 7 Oct 2009 A Low Density Lattice Decoder via Non-parametric Belief Propagation Danny Bickson IBM Haifa Research Lab Mount Carmel, Haifa 31905, Israel Email: danny.bickson@gmail.com Alexander T. Ihler Bren School of Information and Computer Science University of California, Irvine Email: ihler@ics.uci.edu Harel Avissar and Danny Dolev School of Computer Science and Engineering Hebrew University of Jerusalem Jerusalem 91904, Israel Email: {harela01,dolev}@cs.huji.ac.il Abstract— The recent work of Sommer, Feder and Shalvi presented a new family of codes called low density lattice codes (LDLC) that can be decoded efficiently and approach the capacity of the AWGN channel. A linear time iterative decoding scheme which is based on a message-passing formulation on a factor graph is given. In the current work we report our theoretical findings regarding the relation between the LDLC decoder and belief propagation. We show that the LDLC decoder is an instance of non-parametric belief propagation and further connect it to the Gaussian belief propagation algorithm. Our new results enable borrowing knowledge from the non-parametric and Gaussian belief propagation domains into the LDLC domain. Specifically, we give more general convergence conditions for convergence of the LDLC decoder (under the same assumptions of the original LDLC convergence analysis). We discuss how to extend the LDLC decoder from Latin square to full rank, non-square matrices. We propose an efficient construction of sparse generator matrix and its matching decoder. We report preliminary experimental results which show our decoder has comparable symbol to error rate compared to the original LDLC decoder. I. INTRODUCTION Lattice codes provide a continuous-alphabet encoding pro- cedure, in which integer-valued information bits are con- verted to positions in Euclidean space. Motivated by the success of low-density parity check (LDPC) codes [1], recent work by Sommer et al. [2] presented low density lattice codes (LDLC). Like LDPC codes, a LDLC code has a sparse decoding matrix which can be decoded efficiently using an iterative message-passing algorithm defined over a factor graph. In the original paper, the lattice codes were limited to Latin squares, and some theoretical results were proven for this special case. The non-parametric belief propagation (NBP) algorithm is an efficient method for approximated inference on con- tinuous graphical models. The NBP algorithm was originally introduced in [3], but has recently been rediscovered indepen- dently in several domains, among them compressive sensing [4], [5] and low density lattice decoding [2], demonstrating very good empirical performance in these systems. In this work, we investigate the theoretical relations be- tween the LDLC decoder and belief propagation, and show it is an instance of the NBP algorithm. This understanding has both theoretical and practical consequences. From the theory point of view we provide a cleaner and more standard derivation of the LDLC update rules, from the graphical models perspective. From the practical side we propose to use the considerable body of research that exists in the NBP domain to allow construction of efficient decoders. We further propose a new family of LDLC codes as well as a new LDLC decoder based on the NBP algorithm . By utilizing sparse generator matrices rather than the sparse parity check matrices used in the original LDLC work, we can obtain a more efficient encoder and decoder. We introduce the theoretical foundations which are the basis of our new decoder and give preliminary experimental results which show our decoder has comparable performance to the LDLC decoder. The structure of this paper is as follows. Section II overviews LDLC codes, belief propagation on factor graph and the LDLC decoder algorithm. Section III rederive the original LDLC algorithm using standard graphical models terminology, and shows it is an instance of the NBP algo- rithm. Section IV presents a new family of LDLC codes as well as our novel decoder. We further discuss the relation to the GaBP algorithm. In Section V we discuss convergence and give more general sufficient conditions for convergence, under the same assumptions used in the original LDLC work. Section VI brings preliminary experimental results of evaluating our NBP decoder vs. the LDLC decoder. We conclude in Section VII. II. BACKGROUND A. Lattices and low-density lattice codes An n-dimensional lattice Λ is defined by a generator matrix G of size n × n. The lattice consists of the discrete set of points x = (x1, x2, …, xn) ∈Rn with x = Gb, where b ∈Zn is the set of all possible integer vectors. A low-density lattice code (LDLC) is a lattice with a non- singular generator matrix G, for which H = G−1 is sparse. It is convenient to assume that det(H) = 1/det(G) = 1. An (n, d) regular LDLC code has an H matrix with constant row and column degree d. In a latin square LDLC, the values of t

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