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We propose a modification of the cost function of the Hopfield model whosesalient features shine in its Taylor expansion and result in more than pairwiseinteractions with alternate signs, suggesting a unified framework for handlingboth with deep learning and network pruning. In our analysis, we heavily relyon the Hamilton-Jacobi correspondence relating the statistical model with amechanical system. In this picture, our model is nothing but the relativisticextension of the original Hopfield model (whose cost function is a quadraticform in the Mattis magnetization which mimics the non-relativistic Hamiltonianfor a free particle). We focus on the low-storage regime and solve the modelanalytically by taking advantage of the mechanical analogy, thus obtaining acomplete characterization of the free energy and the associatedself-consistency equations in the thermodynamic limit. On the numerical side,we test the performances of our proposal with MC simulations, showing that thestability of spurious states (limiting the capabilities of the standard Hebbianconstruction) is sensibly reduced due to presence of unlearning contributionsin this extended framework.

A relativistic extension of Hopfield neural networks via the mechanical analogy