Multiplicative contexts in associative memories

A system of networks, consisting of a first net that constructs the Kronecker product between two vectors and then sends it to a second net that sustains a correlation memory, defines a context-dependent associative memory. In the real nervous system of higher mammals, the anatomy of the neural connections surely exhibits a considerable amount of local imprecision superimposed on a regular global layout. In order to evaluate the potentialities of the multiplicative devices to constitute plausible biological models, we analyse the performances of a context-dependent memory when the multiplicative net, responsible of the construction of the Kronecker product, presents an incomplete connectivity. Our study shows that a large dimensional system is able to support a considerable amount of incompleteness in the connectivity without a great deterioration of the memory. We establish a scaling relationship between the degree of incompleteness, the capacity of the memory, and the tolerance threshold to imperfections in the output. We then analyse some performances that show the versatility of this kind of network to represent a variety of functions. These functions include a context-modulated novelty filter, a network that computes logical modalities and an adaptive searching device.