  We propose a modification of the cost function of the Hopfield model whose
salient features shine in its Taylor expansion and result in more than pairwise
interactions with alternate signs, suggesting a unified framework for handling
both with deep learning and network pruning. In our analysis, we heavily rely
on the Hamilton-Jacobi correspondence relating the statistical model with a
mechanical system. In this picture, our model is nothing but the relativistic
extension of the original Hopfield model (whose cost function is a quadratic
form in the Mattis magnetization which mimics the non-relativistic Hamiltonian
for a free particle). We focus on the low-storage regime and solve the model
analytically by taking advantage of the mechanical analogy, thus obtaining a
complete characterization of the free energy and the associated
self-consistency equations in the thermodynamic limit. On the numerical side,
we test the performances of our proposal with MC simulations, showing that the
stability of spurious states (limiting the capabilities of the standard Hebbian
construction) is sensibly reduced due to presence of unlearning contributions
in this extended framework.
