SAPA: Sparse Affine Projection Algorithm in ADMM-LP Decoding of LDPC Codes

We present a simplified alternating direction method of multipliers with linear programming (ADMM-LP) decoder for LDPC codes by developing a method of approximate projection onto the parity polytope. The algorithm projects onto the affine hull of the
$X$
closest local codewords for each check of degree d, where
$X$
can be much smaller than d. We name this SAPA, the “sparse affine projection algorithm”. In contrast to exact projection, SAPA does not require a water-filling step and thus can be implemented with lower per-iteration complexity. We present numerical results which demonstrate not only that SAPA's performance, when used as part of an overall decoder, is close to that of exact projection, but also that the use of SAPA does not incur many additional iterations. This is in contrast to other approximate algorithms. Thus, SAPA is appropriate for use in limited-iteration decoders in high-throughput applications. Moreover, we analyze sparsity of polytope projections in the exact ADMM-LP decoding to explain the suitability of the proposed sparse approximation.