2 research outputs found
Applications of correlation inequalities to low density graphical codes
This contribution is based on the contents of a talk delivered at the
Next-SigmaPhi conference held in Crete in August 2005. It is adressed to an
audience of physicists with diverse horizons and does not assume any background
in communications theory. Capacity approaching error correcting codes for
channel communication known as Low Density Parity Check (LDPC) codes have
attracted considerable attention from coding theorists in the last decade.
Surprisingly strong connections with the theory of diluted spin glasses have
been discovered. In this work we elucidate one new connection, namely that a
class of correlation inequalities valid for gaussian spin glasses can be
applied to the theoretical analysis of LDPC codes. This allows for a rigorous
comparison between the so called (optimal) maximum a posteriori and the
computationaly efficient belief propagation decoders. The main ideas of the
proofs are explained and we refer to recent works for the more lengthy
technical details.Comment: 11 pages, 3 figure
The adaptive interpolation method for proving replica formulas. Applications to the Curie-Weiss and Wigner spike models
In this contribution we give a pedagogic introduction to the newly introduced
adaptive interpolation method to prove in a simple and unified way replica
formulas for Bayesian optimal inference problems. Many aspects of this method
can already be explained at the level of the simple Curie-Weiss spin system.
This provides a new method of solution for this model which does not appear to
be known. We then generalize this analysis to a paradigmatic inference problem,
namely rank-one matrix estimation, also refered to as the Wigner spike model in
statistics. We give many pointers to the recent literature where the method has
been succesfully applied