Exploiting Parity-Polytope Geometry in Approximate and Randomized Scheduled ADMM-LP Decoding

We present two strategies to reduce the complexity of the alternating direction method of multipliers when applied to linear programming (ADMM-LP) decoding of low-density parity-check codes. First, to address the high complexity of computing a projection onto the parity polytope, the complexity bottleneck of ADMM-LP decoding, we propose the sparse affine projection algorithm (SAPA). SAPA projects onto the affine hull of \chi \leq d nearby local codewords where the check degree is d and where \chi can be significantly smaller than d. Unlike exact projection, SAPA does not require a water-filling process, and thus can be implemented with lower per-iteration complexity. Second, to reduce the number of effective iterations needed for ADMM-LP decoding, we propose a randomized layered scheduling framework. Rather than updating checks in round-robin fashion in each iteration, more "problematic" checks have a higher probability of being updated. The probability mass function that governs the selection of which checks to update is based upon the location of replica vectors inside (or on) the parity polytope. The resultant decoder converges significantly faster under this randomized scheduling than under round-robin scheduling. This makes it well suited for use in applications that limit the number of iterations.