Pattern disentanglement and reconstruction by modular Hebbian networks

We address the problem of reconstructing a set of K binary patterns Ξ=(ξ1,..,ξK)∈{-1,+1}N×K, starting from their covariance matrix ΞΞT and from a set of mixtures of the form σM=sign(∑μ∈SM⊆{1,..,K}ξμ), with SM=M. To this aim, we engage a modular Hebbian network, specified by the interaction matrix JH=1NΞΞT, where σM is taken as the initial state, repeated in each of the constituting L modules and the stable state resulting from a Gibbs evolution rule is taken as an L-tuple of candidate patterns. Finally, by properly handling JH we can derive the projector matrix JK=Ξ(ΞTΞ)-1ΞT by which we can determine whether these candidate patterns are a good estimate for the ground patterns.