Electrical Engineering
      and Computer Sciences

Electrical Engineering and Computer Sciences


UC Berkeley


2009 Research Summary

Probabilistic Analysis of Linear Programming Decoding

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Konstantinos Daskalakis, Georgios Alexandros Dimakis, Richard M. Karp and Martin Wainwright

National Science Foundation CCF-0515259, National Science Foundation DMS-0528488, National Science Foundation CCF-0515259, Marvell and California MICRO

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 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 flow on hypergraphs. 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.

C. Daskalakis, A. G. Dimakis, R. Karp, and M. J. Wainwright, "Probabilistic Analysis of Linear Programming Decoding," SIAM Symp. Discrete Algorithms (SODA), New Orleans, LA, January 2007.