Pattern disentanglement and reconstruction by modular Hebbian networks

We address the problem of reconstructing a set of K binary patterns Ξ = (ξ ¹,..., ξ K) ∈ \-1, +1 × K Ξ = ( ξ 1 , . . . , ξ K ) ∈ { - 1 , + 1 } N × K , starting from their covariance matrix Ξ Ξ T Ξ Ξ T and from a set of mixtures of the form σ M= sign(∑ _μ ∈ SM ⊆ \1,..., K\ ξ μ) σ M = sign ( ∑ μ ∈ S M ⊆ { 1 , . . . , K } ξ μ ) , with SM =M S M = M . To this aim, we engage a modular Hebbian network, specified by the interaction matrix JH = 1/NΞ Ξ T J H = 1 N Ξ Ξ T , where σ M σ M 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 J H