Probabilistic Analysis of Linear Programming Decoding
<|reference_start|>Probabilistic Analysis of Linear Programming Decoding: 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.<|reference_end|>
@article{daskalakis2007probabilistic,
    title={Probabilistic Analysis of Linear Programming Decoding},
    author={Constantinos Daskalakis, Alexandros G. Dimakis, Richard M. Karp,
  Martin J. Wainwright},
    journal={arXiv preprint arXiv:cs/0702014},
    year={2007},
    doi={10.1109/TIT.2008.926452},
    archivePrefix={arXiv},
    eprint={cs/0702014},
    primaryClass={cs.IT cs.DM math.IT}
}