Guessing Facets: Polytope Structure and Improved LP Decoder
Dimakis, A.G.
Gohari, A.A.
Wainwright, M.J.
Dept. of Electr. Eng. & Comput. Sci., Univ. of California, Berkeley, CA, USA;
This paper appears in: Information Theory, IEEE Transactions on
Publication Date: Aug. 2009
Volume: 55,
Issue: 8
On page(s): 3479-3487
ISSN: 0018-9448
INSPEC Accession Number: 10763991
Digital Object Identifier: 10.1109/TIT.2009.2023735
Current Version Published: 2009-07-14
Abstract
We investigate the structure of the polytope underlying the linear programming (LP) decoder introduced by Feldman, Karger, and Wainwright. We first show that for expander codes, every fractional pseudocodeword always has at least a constant fraction of nonintegral bits. We then prove that for expander codes, the active set of any fractional pseudocodeword is smaller by a constant fraction than that of any codeword. We further exploit these geometrical properties to devise an improved decoding algorithm with the same order of complexity as LP decoding that provably performs better. The method is very simple: it first applies ordinary LP decoding, and when it fails, it proceeds by guessing facets of the polytope, and then resolving the linear program on these facets. While the LP decoder succeeds only if the ML codeword has the highest likelihood over all pseudocodewords, we prove that the proposed algorithm, when applied to suitable expander codes, succeeds unless there exists a certain number of pseudocodewords, all adjacent to the ML codeword on the LP decoding polytope, and with higher likelihood than the ML codeword. We then describe an extended algorithm, still with polynomial complexity, that succeeds as long as there are at most polynomially many pseudocodewords above the ML codeword.
Index
Terms
Available to subscribers and IEEE members.
References
Available to subscribers and IEEE members.
Citing Documents
Available to subscribers and IEEE members.
Cart
