A hierarchy of relaxation between the continuous and convex hull representations
SIAM Journal on Discrete Mathematics
Decoding turbo-like codes via linear programming
Journal of Computer and System Sciences - Special issue on FOCS 2002
Explicit constructions for compressed sensing of sparse signals
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Almost Euclidean subspaces of ℓN1 via expander codes
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
IEEE Transactions on Information Theory - Part 1
The capacity of low-density parity-check codes under message-passing decoding
IEEE Transactions on Information Theory
Using linear programming to Decode Binary linear codes
IEEE Transactions on Information Theory
Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?
IEEE Transactions on Information Theory
LP Decoding Corrects a Constant Fraction of Errors
IEEE Transactions on Information Theory
Probabilistic Analysis of Linear Programming Decoding
IEEE Transactions on Information Theory
LP decoding meets LP decoding: a connection between channel coding and compressed sensing
Allerton'09 Proceedings of the 47th annual Allerton conference on Communication, control, and computing
LP decoding of codes with expansion parameter above 2/3
Information Processing Letters
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Linear programming decoding for low-density parity check codes (and related domains such as compressed sensing) has received increased attention over recent years because of its practical performance --coming close to that of iterative decoding algorithms--- and its amenability to finite-blocklength analysis. Several works starting with the work of Feldman et al. showed how to analyze LP decoding using properties of expander graphs. This line of analysis works for only low error rates, about a couple of orders of magnitude lower than the empirically observed performance. It is possible to do better for the case of random noise, as shown by Daskalakis et al. and Koetter and Vontobel. Building on work of Koetter and Vontobel, we obtain a novel understanding of LP decoding, which allows us to establish a 0.05-fraction of correctable errors for rate-1/2 codes; this comes very close to the performance of iterative decoders and is significantly higher than the best previously noted correctable bit error rate for LP decoding. Unlike other techniques, our analysis directly works with the primal linear program and exploits an explicit connection between LP decoding and message passing algorithms. An interesting byproduct of our method is a notion of a "locally optimal" solution that we show to always be globally optimal (i.e., it is the nearest codeword). Such a solution can in fact be found in near-linear time by a "re-weighted" version of the min-sum algorithm, obviating the need for linear programming. Our analysis implies, in particular, that this re-weighted version of the min-sum decoder corrects up to a 0.05-fraction of errors.