LPDecoding_allerton03 - LP Decoding Jon Feldman Industrial...

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LP Decoding Jon Feldman * Industrial Engineering and Operations Research Columbia University, New York, NY, 10027 jonfeld@ieor.columbia.edu David R. Karger Laboratory for Computer Science MIT, Cambridge, MA, 02139 karger@theory.lcs.mit.edu Martin J. Wainwright Electrical Engineering and Computer Science UC Berkeley, CA, 94720 wainwrig@eecs.berkeley.edu Abstract. Linear programming (LP) relaxation is a common technique used to find good solutions to complex optimization problems. We present the method of “LP decoding”: applying LP relaxation to the problem of maximum-likelihood (ML) decoding. An arbitrary binary-input memoryless channel is considered. This treatment of the LP decoding method places our previous work on turbo codes [6] and low-density parity-check (LDPC) codes [8] into a generic framework. We define the notion of a proper relaxation, and show that any LP decoder that uses a proper relaxation exhibits many useful properties. We describe the notion of pseudocodewords under LP decoding, unifying many known characterizations for specific codes and channels. The fractional distance of an LP decoder is defined, and it is shown that LP decoders correct a number of errors equal to half the fractional distance. We also discuss the application of LP decoding to binary linear codes. We define the notion of a relaxation being symmetric for a binary linear code. We show that if a relaxation is symmetric, one may assume that the all-zeros codeword is transmitted. 1 Introduction The problem of maximum-likelihood (ML) decoding is to find the codeword most likely to have been transmitted, given a corrupted codeword from a noisy channel. Linear programming is the problem of finding an optimal solution to a system of linear inequalities under a linear objective function [2]. In this paper, we consider linear programming (LP) formulations of the ML decoding problem on binary codes. We use LP variables to represent code bits, and the LP objective function is defined by the channel likelihood ratios. Previous work on LP decoding [6, 4, 7, 8, 5] has focused on two specific cases: turbo codes [1] and low-density parity-check codes [11]. These two families of codes have received a lot of attention recently due to their excellent performance. Performance bounds for LP decoding in these cases are for specific LP formulations, code constructions, and/or channel models. In this paper we consider LP decoders for arbitrary binary codes, under an arbitrary binary- input memoryless channel. We provide a framework for designing LP decoders, and general techniques for analyzing them. Central to every LP decoder is its associated polytope : the set of points that satisfy the constraints of the LP. A decoding polytope should contain ev- ery codeword, and should also exclude every binary word that is not a codeword. We define such polytopes as proper . We show that LP decoders that use proper polytopes have the ML certificate property: whenever they output a codeword, it is guaranteed to be the ML codeword. In general, for any sub-optimal decoder, the
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This note was uploaded on 09/05/2009 for the course EECS 227A taught by Professor Martinwainwright during the Fall '04 term at University of California, Berkeley.

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LPDecoding_allerton03 - LP Decoding Jon Feldman Industrial...

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