A factor-graph-based random walk, and its relevance for LP decoding analysis and Bethe entropy characterization

Abstract

Although min-sum algorithm (MSA) and linear programming (LP) decoding are tightly related, it is not straightforward to translate MSA decoding performance analysis techniques to the LP decoding setup. Towards closing this performance analysis techniques gap, Koetter and Vontobel [ITAW 2006] showed how the collection of messages from several MSA decoding iterations can be used to construct a dual witness for LP decoding, thereby deriving some performance results for LP decoding. In a recent breakthrough paper by Arora, Daskalakis, and Steurer (ADS) [STOC 2009], the understanding of the performance of LP decoding was brought to a new level, not only from the perspective of available analysis tools but also from the perspective of significantly improving the known asymptotic LP decoding threshold bounds. ADS achieved this by showing how MSA decoding analysis type results can be used in the primal domain of the LP decoder, along the way also giving evidence that the above detour over the dual domain is neither necessary nor simpler. In the present paper we focus on the geometrical aspects of the ADS paper and show that one of the key results of the ADS paper can be reformulated as the construction of a rather nontrivial class of supersets of the fundamental cone, where these supersets are convex cones that are generated by vectors that are derived from computation trees and minimal valid deviations therein. As we will discuss, the main ingredient that allows the verification of this superset construction is a certain class of backtrackless random walks on the code's normal factor graph. Moreover, formulating our results in terms of normal factor graphs will facilitate the generalization of the geometrical results of the ADS paper to setups with non-uniform node degrees, with other types of constraint function nodes, and with no restrictions on the girth. We conclude the paper by showing connections between the entropy rates of the above-mentioned random walks and the Bethe entropy function of the normal factor graph that these random walks are defined on.