Information Geometry for Decoding

Abstract

We revisit the idea of using elements of information geometry for decoding low-density parity-check (LDPC) codes, as introduced by Ikeda et al. In this work, we explicitly compute the m-projection to an e-flat submanifold, in the case of a binary symmetric channel and the Gaussian channel. We exemplify the algorithm by testing moderate size Gallager codes. To approach decoding problems, we show general theorems based on alternating projections in the framework of information geometry, inspired by von Neumann’s theorem for the convergence of alternating projections in Hilbert spaces. More precisely, consider the manifold S of the probability distributions on the n-dimensional hypercube (i.e., the set of binary sequences of length n). Let p be in S. In the case of two intersecting m-flat or e-flat submanifolds, the method of alternating projections on the two submanifolds converges to the projection of p on their intersection. This result is also generalized to a finite family of submanifolds of S.