Title: Neural Networks retrieving Boolean patterns in a sea of Gaussian ones,
Abstract: Restricted Boltzmann Machines are key tools in Machine Learning and are
described by the energy function of bipartite spin-glasses. From a statistical
mechanical perspective, they share the same Gibbs measure of Hopfield networks
for associative memory. In this equivalence, weights in the former play as
patterns in the latter. As Boltzmann machines usually require real weights to
be trained with gradient descent like methods, while Hopfield networks
typically store binary patterns to be able to retrieve, the investigation of a
mixed Hebbian network, equipped with both real (e.g., Gaussian) and discrete
(e.g., Boolean) patterns naturally arises. We prove that, in the challenging
regime of a high storage of real patterns, where retrieval is forbidden, an
extra load of Boolean patterns can still be retrieved, as long as the ratio
among the overall load and the network size does not exceed a critical
threshold, that turns out to be the same of the standard
Amit-Gutfreund-Sompolinsky theory. Assuming replica symmetry, we study the case
of a low load of Boolean patterns combining the stochastic stability and
Hamilton-Jacobi interpolating techniques. The result can be extended to the
high load by a non rigorous but standard replica computation argument.