Decomposition Methods for Large Scale LP Decoding

When binary linear error-correcting codes are used over symmetric channels, a relaxed version of the maximum likelihood decoding problem can be stated as a linear program (LP). This LP decoder can be used to decode error-correcting codes at bit-error-rates comparable to state-of-the-art belief propagation (BP) decoders, but with significantly stronger theoretical guarantees. However, LP decoding when implemented with standard LP solvers does not easily scale to the block lengths of modern error correcting codes. In this paper we draw on decomposition methods from optimization theory, specifically the Alternating Directions Method of Multipliers (ADMM), to develop efficient distributed algorithms for LP decoding. The key enabling technical result is a "two-slice" characterization of the geometry of the parity polytope, which is the convex hull of all codewords of a single parity check code. This new characterization simplifies the representation of points in the polytope. Using this simplification, we develop an efficient algorithm for Euclidean norm projection onto the parity polytope. This projection is required by ADMM and allows us to use LP decoding, with all its theoretical guarantees, to decode large-scale error correcting codes efficiently. We present numerical results for LDPC codes of lengths more than 1000. The waterfall region of LP decoding is seen to initiate at a slightly higher signal-to-noise ratio than for sum-product BP, however an error floor is not observed for LP decoding, which is not the case for BP. Our implementation of LP decoding using ADMM executes as fast as our baseline sum-product BP decoder, is fully parallelizable, and can be seen to implement a type of message-passing with a particularly simple schedule.Comment: 35 pages, 11 figures. An early version of this work appeared at the 49th Annual Allerton Conference, September 2011. This version to appear in IEEE Transactions on Information Theor

ADMM and Reproducing Sum-Product Decoding Algorithm Applied to QC-MDPC Code-based McEliece Cryptosystems

QC-MDPC (quasi cyclic moderate density parity check) code-based McEliece cryptosystems are considered to be one of the candidates for post-quantum cryptography. Decreasing DER (decoding error rate) is one of important factor for their security, since recent attacks to these cryptosystems effectively use DER information. In this paper, we pursue the possibility of optimization-base decoding, concretely we examine ADMM (alternating direction method of multipliers), a recent developing method in optimization theory. Further, RSPA (reproducing sum-product algorithm), which efficiently reuse outputs of SPA (sum-product algorithm) is proposed for the reduction of execution time in decoding. By numerical simulations, we show that the proposing scheme shows considerable decrement in DER compared to the conventional decoding methods such as BF (bit-flipping algorithm) variants or SPA

Extracting Signals and Graphical Models from Compressed Measurements

The thesis is to give an integrated approach to efficiently learn the interdependency relation among high dimensional signal components and reconstruct signals from observations collected in a linear sensing system, Broadly speaking, the research topics consists of three parts: (i) interdependency relation learning; (ii) sensing system design; and (iii) signal reconstruction. In the interdependency relation learning part, we considered both the parametric and non-parametric methods to learn the graphical structure under the noisy indirect measurements. In the sensing system design part, we introduced a density evolution framework to design sensing systems for compressive sensing for the first time. In the signal reconstruction part, we focused on the signal reconstruction with a given sensing system, which consists of three parts: signal reconstruction with inexact knowledge of the sensing system; signal reconstruction with the signal being contaminated by undesired noise; signal reconstruction with the signal belonging to a union of convex sets.Ph.D