Probabilistic analysis of linear programming decoding

  • Authors:
  • Constantinos Daskalakis;Alexandros G. Dimakis;Richard M. Karp;Martin J. Wainwright

  • Affiliations:
  • UC Berkeley;UC Berkeley;UC Berkeley;UC Berkeley

  • Venue:
  • SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
  • Year:
  • 2007

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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 Feld-man et al. succeeds in correcting a constant fraction of errors with high probability. The fraction of correctable errors guaranteed by our analysis surpasses all prior 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 "almost expansion" in random bipartite graphs, in which one requires only that almost every (as opposed to every) set of a certain size expands, with expansion coefficients much larger than the classical case.