| cs/0702014 (February 2007) | | Probabilistic Analysis of Linear Programming Decoding | | Constantinos Daskalakis, Alexandros G. Dimakis, Richard M. Karp and Martin J. Wainwright | | Received. 02 February 2007 Last updated. 10 March 2008 | | Abstract. We initiate the probabilistic analysis of linear programming (LP) decoding of low-density parity-check (LDPC) codes. Specifically, we show that for a random LDPC code ensemble, the linear programming decoder of Feldman et al. succeeds in correcting a constant fraction of errors with high probability. The fraction of correctable errors guaranteed by our analysis surpasses previous non-asymptotic results for LDPC codes, and in particular exceeds the best previous finite-length result on LP decoding by a factor greater than ten. This improvement stems in part from our analysis of probabilistic bit-flipping channels, as opposed to adversarial channels. At the core of our analysis is a novel combinatorial characterization of LP decoding success, based on the notion of a generalized matching. An interesting by-product of our analysis is to establish the existence of "probabilistic expansion" in random bipartite graphs, in which one requires only that almost every (as opposed to every) set of a certain size expands, for sets much larger than in the classical worst-case setting. | | Categories. cs.IT cs.DM math.IT | | Subject. Information Theory; Discrete Mathematics; | | Comment. To appear, IEEE Transactions on Information Theory, (replaces shorter version that appeared in SODA'07) |
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