A quantitative analysis of a generalized Hopfield model that stores and retrieves mismatched memory patterns

Abstract

We study a class of Hopfield models where the memories are represented by a mixture of Gaussian and binary variables and the neurons are Ising spins. We study the properties of this family of models as the relative weight of the two kinds of variables in the patterns varies. We quantitatively determine how the retrieval phase squeezes toward zero as the memory patterns contain a larger fraction of mismatched variables. As the memory is purely Gaussian retrieval is lost for any positive storage capacity. It is shown that this comes about because of the spherical symmetry of the free energy in the Gaussian case. Introducing two different memory pattern overlaps between spin configurations and each contribution to the pattern from the two kinds of variables one can observe that the Gaussian parts of the patterns act as a noise, making retrieval more difficult. The basins of attraction of the states, the accuracy of the retrieval and the storage capacity are studied by means of Monte Carlo numerical simulations. We uncover that even in the limit where the network capacity shrinks to zero, the (few) retrieval states maintain a large basin of attraction and large overlaps with the mismatched patterns. So the network can be used for retrieval, but with a very small capacity.

Thermodynamics of bidirectional associative memories

“…In figure 7 we further notice how, even beyond the predicted critical capacity, the magnetization does not fall to 0: a first explanation is that finite-size effect can result into the dynamics being stuck in spurious attractors characterized by a partial overlap with respect to more than one pattern (mixed states), that are not present in the MF theory in this regime, a phenomenon already observed in [8] for the Hopfield model. Secondly, this phenomenon can be understood also in terms of the lack of patterns' orthogonality at finite size, and it would would explain why in both figures 7 (a2)-( b2)-( a3)-(b3) (i.e.…”

“…by setting z = u + iv,z † = u − iv: the resulting object is a simple Gaussian integral over the real plane. Thus, equation ( 13) is used to decouple the interaction terms in the partition functions equation (8). introducing a pair of complex conjugate variables for each pattern µ.…”

“…In addition, extensions to different support for the spin variables and/or the patterns (e.g. real-valued spins and/or pattern components) have been deeply analyzed [3,[7][8][9].…”