Statistical mechanics of an error correcting code using monotonic and nonmonotonic treelike multilayer perceptrons

An error correcting code using a treelike multilayer perceptron is proposed. An original message ${\mathit{s}}^0$ is encoded into a codeword ${\mathit{y}}_0$ using a treelike committee machine (committee tree) or a treelike parity machine (parity tree). Based on these architectures, several schemes featuring monotonic or nonmonotonic units are introduced. The codeword ${\mathit{y}}_0$ is then transmitted via a binary asymmetric channel where it is corrupted by noise. The analytical performance of these schemes is investigated using the replica method of statistical mechanics. Under some specific conditions, some of the proposed schemes are shown to saturate the Shannon bound at the infinite codeword length limit. The influence of the monotonicity of the units on the performance is also discussed.

Figure 1

The binary asymmetric channel (BAC).

Figure 2

Layout of the scheme.

Figure 3

Left: general architecture of the treelike multilayer perceptrons with $N$ input units and $K$ hidden units. Right: the nonmonotonic function $f_k$.