Sanjeev Arora
Constantinos Daskalakis
David Steurer
ABSTRACT
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.
- E. J. Candes and T. Tao. Near--optimal signal recovery from random projections: Universal encoding strategies? IEEE Transactions on Information Theory, 52(12):5406--5425, 2006. Google Scholar
Digital Library
- J. Chen, A. Dholakia, E. Eleftheriou, M. Fossorier, and X.-Y. Hu. Reduced-complexity decoding of LDPC codes. Communications, IEEE Transactions on, 53(8):1288--1299, Aug. 2005.Google ScholarCross Ref
- J. Chen and M. Fossorier. Density evolution for two improved BP-based decoding algorithms of LDPC codes. Communications Letters, IEEE, 6(5):208--210, May 2002.Google ScholarCross Ref
- J. Chen and M. Fossorier. Near optimum universal belief propagation based decoding of low-density parity check codes. Communications, IEEE Transactions on, 50(3):406--414, Mar 2002.Google ScholarCross Ref
- C. Daskalakis, A. G. Dimakis, R. M. Karp, and M. J. Wainwright. Probabilistic analysis of linear programming decoding. IEEE Transactions on Information Theory, 54(8):3565--3578, 2008. Google ScholarDigital Library
- J. Feldman and D. R. Karger. Decoding turbo-like codes via linear programming. J. Comput. System Sci., 68(4):733--752, 2004. Google ScholarDigital Library
- J. Feldman, T. Malkin, R. A. Servedio, C. Stein, and M. J. Wainwright. Lp decoding corrects a constant fraction of errors. IEEE Transactions on Information Theory, 53(1):82--89, 2007. Google ScholarDigital Library
- J. Feldman, M. J. Wainwright, and D. R. Karger. Using linear programming to decode binary linear codes. IEEE Trans. Inform. Theory, 51(3):954--972, 2005. Google ScholarDigital Library
- B. J. Frey and R. Koetter. The attenuated max-product algorithm. In Advanced mean field methods (Birmingham, 1999), Neural Inf. Process. Ser., pages 213--227. MIT Press, Cambridge, MA, 2001.Google Scholar
- R. G. Gallager. Low-density parity check codes. MIT Press, Cambridge, MA, 1963.Google ScholarCross Ref
- V. Guruswami, J. R. Lee, and A. A. Razborov. Almost euclidean subspaces of ln 1 via expander codes. In SODA, pages 353--362, 2008. Google ScholarDigital Library
- P. Indyk. Explicit constructions for compressed sensing of sparse signals. In SODA, pages 30--33, 2008. Google ScholarDigital Library
- R. Koetter and P. O. Vontobel. On the block error probability of LP decoding of LDPC codes. In Inaugural Workshop of the Center for Information Theory and its Applications, 2006. Available as eprint arXiv:cs/0602086v1.Google Scholar
- T. Richardson and R. Urbanke. The capacity of low-density parity-check codes under message-passing decoding. IEEE Trans. on Info. Theory, 47(2), Feb. 2001. Google ScholarDigital Library
- H. D. Sherali and W. P. Adams. A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J. Discrete Math., 3(3):411--430, 1990. Google ScholarDigital Library
- A. Shokrollahi. Ldpc codes: An introduction. M. Sipser and D. Spielman. Expander codes. IEEE Trans. Info. Theory, 42:1710--1722, November 1996. Google ScholarCross Ref
- N. Wiberg. Codes and Decoding on General Graphs. PhD thesis, Linkoeping University, Sweden, 1996.Google Scholar
Index Terms
- Message passing algorithms and improved LP decoding
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